IJMMS 2003:66, 4145–4182
PII. S0161171203303370
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS
IN THE CALCULUS OF VARIATIONS
G. J. ZALMAI
Received 20 March 2003
Parametric and nonparametric necessary and sufficient optimality conditions are
established for a class of nonconvex variational problems with generalized fractional objective functions and nonlinear inequality constraints containing arbitrary norms. Based on these optimality criteria, ten parametric and parameterfree dual problems are constructed and appropriate duality theorems are proved.
These optimality and duality results contain, as special cases, similar results for
minmax fractional variational problems involving square roots of positive semidefinite quadratic forms as well as for variational problems with fractional, discrete
max, and conventional objective functions, which are particular cases of the main
problem considered in this paper. The duality models presented here subsume
various existing duality formulations for variational problems and include variational generalizations of a great variety of cognate dual problems investigated
previously in the area of finite-dimensional nonlinear programming by an assortment of ad hoc methods.
2000 Mathematics Subject Classification: 49K35, 49N15, 90C32, 90C47 .
1. Introduction. In this paper, we establish necessary and sufficient optimality conditions and construct a fairly large number of parametric and
parameter-free duality models for the following unorthodox variational problem:
(P)
b a fi t, x(t), ẋ(t) + Ai (t)x(t) L(i) dt
Minimize max b 1≤i≤k
a gi t, x(t), ẋ(t) − Bi (t)x(t) M(i) dt
(1.1)
subject to
hj t, x(t), ẋ(t) + Cj (t)x(t)N(j) ≤ 0,
x ∈ PWSn [a, b],
t ∈ [a, b], j ∈ m,
(1.2)
(1.3)
where PWSn [a, b] is the space of all piecewise smooth n-dimensional vector
functions x defined on the compact interval [a, b] of the real line R, with the
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G. J. ZALMAI
norm x = x∞ + Dx∞ , where the differentiation operator D is given by
t
y = Dx ⇐⇒ x(t) = x(a) +
y(s)ds;
(1.4)
a
thus D = d/dt except at discontinuities, x a and x b are given vectors in Rn
(n-dimensional Euclidean space), ẋ(t) = dx(t)/dt; fi , gi , i ∈ k ≡ {1, 2, . . . , k},
and hj , j ∈ m, are continuously differentiable real-valued functions defined on
[a, b] × Rn × Rn ; Ai (t), Bi (t), i ∈ k, and Cj (t), j ∈ m, are, respectively, pi × n,
qi × n, and rj × n matrices whose entries are continuous real-valued functions
defined on [a, b]; · L(i) , · M(i) , i ∈ k, and · N(j) , j ∈ m, are arbitrary
norms, and, for each i ∈ k,
b
fi t, x(t), ẋ(t) + Ai (t)x(t)L(i) dt ≥ 0,
a
b
a
gi t, x(t), ẋ(t) − Bi (t)x(t)M(i) dt > 0,
(1.5)
for all x satisfying the constraints of (P).
Finite-dimensional counterparts of (P) are known as generalized fractional
programming problems in the literature of mathematical programming. These
problems have arisen in the areas of multiobjective programming [1], approximation theory [2, 3, 12, 16], goal programming [5, 11], and economics [15]
among others.
The notion of duality for generalized linear fractional programming was initially considered by von Neumann [15] in his investigation of economic equilibrium problems. More recently, various optimality criteria, duality formulations, and computational algorithms for several classes of generalized linear
and nonlinear fractional programming problems have appeared in the related
literature. A fairly extensive list of references pertaining to various aspects of
these problems is given in [20].
In contrast to the finite-dimensional case, infinite-dimensional problems of
this type and, in particular, variational problems with generalized fractional
objective functions have not yet received much attention in the literature of
optimization theory and, consequently, at the present no significant results of
any kind are available for these problems.
In the present study, we will establish, under suitable convexity assumptions, both parametric and nonparametric necessary and sufficient optimality
conditions, construct several parametric and parameter-free duality models,
and prove appropriate duality theorems. Our approach for achieving these
goals is based on a set of necessary optimality conditions for a related problem discussed in [4] and two ancillary problems that are intimately linked to
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4147
(P). These problems will enable us to treat (P) within the framework of convex
programming. As pointed out earlier, the optimality and duality results established in the present study improve and extend a number of similar existing
results for variational problems and provide continuous analogues of many
cognate results previously obtained in the area of nonlinear programming. In
particular, they generalize the results of [17] and are closely related to those
given in [18, 19].
The rest of this paper is organized as follows. In Section 2 we recall a set of
necessary optimality conditions given in [4] for a special case of (P). In Section 3
we utilize these optimality conditions in conjunction with some other auxiliary results to establish both parametric and nonparametric necessary optimality principles for (P). We begin our discussion of duality for (P) in Section 4
where we introduce two parametric duality models and prove weak, strong, and
strict converse duality theorems under appropriate convexity assumptions. In
Sections 5 and 6 we formulate a total of eight parameter-free duality models
for (P) and prove appropriate duality theorems. Finally, in Section 7 we briefly
discuss an important special case of (P) which involves square roots of positive
semidefinite quadratic forms.
It is evident that all the results obtained for (P) are also applicable, when appropriately specialized, to the following classes of variational problems with
fractional, discrete max, and conventional objective functions, which are particular cases of (P):
(P1)
b a f1 t, x(t), ẋ(t) + A1 (t)x(t) L(1) dt
Minimize b ,
x∈F
a g1 t, x(t), ẋ(t) − B1 (t)x(t) M(1) dt
(1.6)
(P2)
b
Minimize max
x∈F
1≤i≤k a
fi t, x(t), ẋ(t) + Ai (t)x(t)L(i) dt,
(1.7)
(P3)
b
Minimize
x∈F
a
f1 t, x(t), ẋ(t) + A1 (t)x(t)L(1) dt,
(1.8)
where F (assumed to be nonempty) is the feasible set of (P), that is,
F = x ∈ PWSn [a, b] : (1.1) and (1.2) hold .
(1.9)
Although different concepts of duality have been discussed for various types
of conventional variational (and optimal control) problems (see, e.g., [9, 13] and
the references therein), constrained variational problems like (P1) and (P2) with
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G. J. ZALMAI
nonstandard objective functions have not received much attention in the area
of optimization theory. In contrast, their static analogues have been studied
extensively during the last three decades. Recent surveys of fractional programming are given in [6, 14], and a fairly extensive bibliography is included
in [14]. Similarly, a detailed account of discrete (and continuous) minmax theory and methods is available in [7].
Evidently, a salient feature of (P) is the presence of arbitrary norms in its
objective and constraint functions. Optimization problems involving norms
occur in many areas of the decision sciences, applied mathematics, and engineering. These problems are encountered most frequently in location theory,
approximation theory, and engineering design. A number of references dealing
with various aspects of these problems are given in [17] (see also [4, 18, 19]).
2. Preliminaries. In our derivation of optimality conditions for (P) in the
next section, we will need an optimality result of [4] for the problem
(P4)
b
Minimize
a
f t, x(t), ẋ(t) + A(t)x(t)L ]dt
(2.1)
subject to (1.1), (1.2), (1.3), and
b
a
Hs t, x(t), ẋ(t) + Es (t)x(t)P (s) dt ≤ 0,
s ∈ M,
(2.2)
where f and Hs , s ∈ M, are continuously differentiable real-valued functions
defined on [a, b] × Rn × Rn ; A(t) and Es (t), s ∈ M, are, respectively, µ × n and
νs ×n matrices whose entries are continuous real-valued functions defined on
[a, b], and · L and · P (s) , s ∈ M, are arbitrary norms.
Constraints of type (2.2) are not explicitly included in the problem treated in
[4]. However, it is easily seen from the abstract reformulation of the problem
and proof of [4, Theorem 1] that such integral inequality constraints can indeed
be incorporated in the problem under consideration without any difficulty.
The following result for (P4) can be deduced from [4, Theorem 1].
Theorem 2.1 [4]. Assume that the functions f (t, ·, ·), hj (t, ·, ·), j ∈ m,
and Hs (t, ·, ·), s ∈ M, are convex on Rn × Rn throughout [a, b] and that the
constraints of (P4) satisfy Slater’s constraint qualification, that is, there exists
x̄ ∈ PWSn [a, b] such that x̄(a) = x a , x̄(b) = x b , and
˙ + Cj (t)x̄ N(j) < 0, t ∈ [a, b], j ∈ m,
hj t, x̄, x̄
b
˙ + Es (t)x̄ P (s) dt < 0, s ∈ M.
Hs t, x̄, x̄
a
(2.3)
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
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Then a feasible solution x ∗ of (P4) is optimal if and only if there exist v ∗ ∈
µ
rj
∗
j
s
∈ RM
PWSm
+ , ζ ∈ PWS [a, b], η ∈ PWS [a, b], j ∈ m, θ ∈
+ [a, b], w
νs
PWS [a, b], s ∈ M, such that the following relations hold for all t ∈ [a, b]:
m
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + Cj (t)T ηj (t)
∇2 f t, x ∗ , ẋ ∗ + A(t)T ζ +
j=1
+
M
ws∗ ∇2 Hs t, x ∗ , ẋ ∗ + Es (t)T θ s (t)
s=1
m
− D ∇3 f t, x ∗ , ẋ ∗ +
vj∗ (t)∇3 hj t, x ∗ , ẋ ∗
j=1
+
ws∗ ∇3 Hs t, x ∗ , ẋ ∗ = 0,
M
s=1
m
vj∗ (t) hj t, x ∗ , ẋ ∗ + Cj (t)x ∗ N(j) = 0,
(2.4)
j=1
b M
a s=1
ζ(t)∗ ≤ 1,
L
ws∗ Hs t, x ∗ , ẋ ∗ + Es (t)x ∗ P (s) dt = 0,
j
η (t)∗
s
θ (t)∗ ≤ 1,
j ∈ m,
P (s)
T
∗
∗
ζ(t) A(t)x = A(t)x L ,
ηj (t)T Cj (t)x ∗ = Cj (t)x ∗ N(j) , j ∈ m,
θ s (t)T Es (t)x ∗ = Es (t)x ∗ P (s) , s ∈ M,
N(j)
≤ 1,
s ∈ M,
m
M
where PWSm
+ [a, b] = {v ∈ PWS [a, b] : v(t) ≥ 0 for all t ∈ [a, b]}, R+ = {w ∈
M
T
R : w ≥ 0}, Q is the transpose of the matrix Q, ∇2 F and ∇3 F denote the
partial gradients of the function F : [a, b] × Rn × Rn → R, (t, x(t), ẋ(t)) →
F (t, x(t), ẋ(t)), with respect to its second and third arguments, respectively,
that is, ∇2 F = (∂F /∂x1 , . . . , ∂F /∂xn )T and ∇3 F = (∂F /∂ ẋ1 , . . . , ∂F /∂ ẋn )T , and
· ∗
J denotes the dual norm to · J .
˙,
In the above theorem, the argument t of the vector-valued functions x̄, x̄
∗
x , and ẋ was omitted for the sake of notational simplicity. This practice will
be continued throughout the sequel.
∗
3. Optimality conditions. In this section, we adopt a Dinkelbach-type [8]
indirect parametric approach for establishing a set of necessary optimality
conditions for (P). The intermediate auxiliary problem making this possible
has the following form:
(Pλ)
b
Minimize max
x∈F
1≤i≤k a
fi t, x, ẋ + Ai (t)x L(i) − λ gi t, x, ẋ − Bi (t)x M(i) dt,
(3.1)
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G. J. ZALMAI
where λ ∈ R+ ≡ [0, ∞) is a parameter. It is well known in the area of generalized fractional programming that this problem is closely related to (P). The
relationship between (P) and (Pλ) needed for our present purposes is stated in
the following lemma whose proof is straightforward and hence omitted.
Lemma 3.1. Let λ∗ be the optimal value of (P) and let v(λ) be the optimal
value of (Pλ) for any λ ∈ R+ such that (Pλ) has an optimal solution. If x ∗ is an
optimal solution of (P), then x ∗ is an optimal solution of (Pλ∗ ) and v(λ∗ ) = 0.
It is clear that (Pλ) is in turn equivalent to the following problem:
(EPλ) Minimize µ subject to x ∈ F, µ ∈ R, and
b
a
fi t, x, ẋ + Ai (t)x L(i) − λ gi t, x, ẋ − Bi (t)x M(i) dt ≤ µ,
i ∈ k.
(3.2)
In view of Lemma 3.1 and the equivalence of (Pλ) and (EPλ), it is evident that
if x ∗ is an optimal solution of (P) with optimal value λ∗ , then (x ∗ , µ ∗ ) =
(x ∗ , 0) is an optimal solution of (EPλ∗ ). We use this observation in the proof
of Theorem 3.2 which is the main result of this section. We first specify our
basic assumptions which will remain in force throughout the sequel.
(a) The functions fi (t, ·, ·), −gi (t, ·, ·), i ∈ k, and hj (t, ·, ·), j ∈ m, are convex
on Rn × Rn throughout [a, b].
(b) The constraints of (P) satisfy Slater’s constraint qualification (see Theorem
2.1).
Theorem 3.2. Let x ∗ ∈ F be an optimal solution of (P). Then there exist
k
m
∗
∗
∗i
∈ PWSpi [a, b], β∗i ∈
u ∈ Rk+ , i=1 u∗
i = 1, λ ∈ R+ , v ∈ PWS+ [a, b], α
qi
rj
∗j
PWS [a, b], i ∈ k, γ ∈ PWS [a, b], j ∈ m, such that the following relations
hold for all t ∈ [a, b]:
∗
k
∗
∗
+ Ai (t)T α∗i (t)
u∗
i ∇2 fi t, x , ẋ
i=1
− λ∗ ∇2 gi t, x ∗ , ẋ ∗ − Bi (t)T β∗i (t)
+
m
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + Cj (t)T γ ∗j (t)
(3.3)
j=1
−D
k
∗
∗
− λ∗ ∇3 gi t, x ∗ , ẋ ∗
u∗
i ∇3 fi t, x , ẋ
i=1
+
m
vj∗ (t)∇3 hj t, x ∗ , ẋ ∗
= 0,
j=1
m
j=1
vj∗ (t) hj t, x ∗ , ẋ ∗ + Cj (t)x ∗ N(j) = 0,
(3.4)
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
b k
a
∗
∗
u∗
+ Ai (t)x ∗ L(i)
i fi t, x , ẋ
i=1
− λ gi t, x ∗ , ẋ ∗ − Bi (t)x ∗ M(i) dt = 0,
∗i ∗
∗j
β (t)
γ (t)∗ ≤ 1,
i ∈ k,
M(i) ≤ 1,
N(j)
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(3.5)
∗
∗i ∗
α (t)
L(i)
≤ 1,
α∗i (t)T Ai (t)x ∗ = Ai (t)x ∗ L(i) ,
β∗i (t)T Bi (t)x ∗ = Bi (t)x ∗ M(i) ,
γ ∗j (t)T Cj (t)x ∗ = Cj (t)x ∗ N(j) ,
j ∈ m.
j ∈ m,
(3.6)
i ∈ k,
(3.7)
(3.8)
Proof. Since x ∗ is an optimal solution of (P), by Lemma 3.1, it is an optimal solution of (Pλ∗ ), where λ∗ is the optimal value of (P). This implies that
(x ∗ , µ ∗ ) = (x ∗ , 0) is an optimal solution of (EPλ∗ ). By hypothesis, there exists
x̄ ∈ PWSn [a, b] with x̄(a) = x a and x̄(b) = x b , at which Slater’s constraint
qualification is satisfied. Because of the special structure of the constraints of
(EPλ∗ ), it is obvious that for some µ̄ ∈ R, Slater’s constraint qualification holds
for (EPλ∗ ) at (x̄, µ̄). Therefore, by Theorem 2.1 (applied to (EPλ∗ )), there exist
u∗ , v ∗ , α∗i , β∗i , i ∈ k, and γ ∗j , j ∈ m, as specified above, such that (3.3), (3.4),
(3.5), (3.6), (3.7), and (3.8) hold for all t ∈ [a, b].
In order to demonstrate that the necessary optimality conditions of Theorem
3.2 are also sufficient for optimality of x ∗ , we need the generalized CauchySchwarz inequality [10]: for each w, z ∈ Rn , one has
w T z ≤ wz∗ .
(3.9)
We also need the following lemma which provides an alternative expression for
the objective function of (P); its proof is straightforward and hence omitted.
Lemma 3.3. For each x ∈ PWSn [a, b],
b a fi t, x(t), ẋ(t) + Ai (t)x(t) L(i) dt
ϕ(x) ≡ max b 1≤i≤k
a gi t, x(t), ẋ(t) − Bi (t)x(t) M(i) dt
i=1 ui fi t, x(t), ẋ(t) + Ai (t)x(t) L(i) dt
a
= max b k
,
u∈U
i=1 ui gi t, x(t), ẋ(t) − Bi (t)x(t) M(i) dt
a
b k
(3.10)
where
U = u ∈ Rk+ :
k
i=1
ui = 1 .
(3.11)
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G. J. ZALMAI
Theorem 3.4. Let x ∗ ∈ F, let λ∗ = ϕ(x ∗ ), and assume that there exist
∗i
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b], i ∈ k, γ ∗j ∈
u ∈ U , v ∗ ∈ PWSm
+ [a, b], α
rj
PWS [a, b], j ∈ m, such that (3.3), (3.4), (3.5), (3.6), (3.7), and (3.8) hold for all
t ∈ [a, b]. Then x ∗ is an optimal solution of (P).
∗
Proof. Let x be an arbitrary feasible solution of (P). Keeping in mind that
u∗ ≥ 0, λ∗ ≥ 0, and v ∗ (t) ≥ 0 for each t ∈ [a, b], we have
b k
a
∗
u∗
gi t, x, ẋ − Bi (t)x M(i) dt
i fi t, x, ẋ + Ai (t)x L(i) − λ
i=1
=
≥
=
b k
a
i=1
b k
a
u∗
i fi t, x, ẋ + Ai (t)x L(i)
− λ∗ gi t, x, ẋ − Bi (t)x M(i)
− fi t, x ∗ , ẋ ∗ − Ai (t)x ∗ L(i)
+ λ∗ gi t, x ∗ , ẋ ∗ − Bi (t)x ∗ M(i) dt
T ∗
∗ T
x − x ∗ + ∇3 fi t, x ∗ , ẋ ∗
ẋ − ẋ ∗
u∗
i ∇2 fi t, x , ẋ
i=1
b
k
a
by (3.5)
T T x − x ∗ + ∇3 gi t, x ∗ , ẋ ∗
ẋ − ẋ ∗
− λ∗ ∇2 gi t, x ∗ , ẋ ∗
+ Ai (t)x L(i) + λ∗ Bi (t)x M(i) − Ai (t)x ∗ L(i)
− λ∗ Bi (t)x ∗ M(i) dt
by the convexity of fi (t, ·, ·) and −gi (t, ·, ·), i ∈ k
u∗
i
T
T ∇3 fi t, x ∗ , ẋ ∗ − λ∗ ∇3 gi t, x ∗ , ẋ ∗
ẋ − ẋ ∗
i=1
−
+ Ai (t)x L(i) + λ∗ Bi (t)x M(i)
− Ai (t)x ∗ L(i) − λ∗ Bi (t)x ∗ M(i)
k
∗i T
∗ ∗i
T
u∗
i α (t) Ai (t) + λ β (t) Bi (t)
i=1
+
m
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + γ ∗j (t)T Cj (t)
j=1
T
T
− D ∇3 fi t, x ∗ , ẋ ∗ − λ∗ ∇3 gi t, x ∗ , ẋ ∗
m
T +
vj∗ (t)∇3 hj t, x ∗ , ẋ ∗
x − x ∗ dt
j=1
≥
b
k
a
i=1
u∗
i
by (3.3)
T
T ∇3 fi t, x ∗ , ẋ ∗ − λ∗ ∇3 gi t, x ∗ , ẋ ∗
ẋ − ẋ ∗
+ Ai (t)x L(i) + λ∗ Bi (t)x M(i)
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
− Ai (t)x ∗ −
k
α∗i (t)∗ Ai (t)x u∗
L(i)
L(i)
i
i=1
∗ + λ∗ β∗i (t)M(i) Bi (t)x M(i)
− α∗i (t)T Ai (t)x ∗ − λ∗ β∗i (t)T Bi (t)x ∗
+
m
+
∗ + γ ∗j (t)N(j) Cj (t)x N(j) − γ ∗j (t)T Cj (t)x ∗
k
T ∗
∗ T
u∗
− λ∗ ∇3 gi t, x ∗ , ẋ ∗
i ∇3 fi t, x , ẋ
i=1
+
m
T vj∗ (t)∇3 hj t, x ∗ , ẋ ∗
ẋ − ẋ ∗ dt
j=1
≥
b m
a
j=1
b m
a
=−
≥0
by (3.9) and integration by parts
T T x − x ∗ + ∇3 hj t, x ∗ , ẋ ∗
ẋ − ẋ ∗
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗
+ Cj (t)x N(j) − γ ∗j (t)T Cj (t)x ∗ dt
by (3.6) and (3.7)
vj∗ (t) hj t, x ∗ , ẋ ∗ − hj (t, x, ẋ)
j=1
b m
a
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗
j=1
≥−
4153
∗
Bi (t)x ∗ M(i)
L(i) − λ
− Cj (t)x N(j) + γ ∗j (t)T Cj (t)x ∗ dt
by the convexity of hj (t, ·, ·), j ∈ m
vj∗ (t) hj (t, x, ẋ) − Cj (t)x N(j) dt
by (3.4) and (3.8)
j=1
(by the feasibility of x).
(3.12)
Now using this inequality and Lemma 3.3, we see that
fi t, x, ẋ + Ai (t)x L(i) dt
ϕ(x) = max b k
u∈U
i=1 ui gi t, x, ẋ − Bi (t)x M(i) dt
a
b k
∗
i=1 ui fi t, x, ẋ(t) + Ai (t)x L(i) dt
a
≥ b k
∗
i=1 ui gi t, x, ẋ − Bi (t)x M(i) dt
a
≥ λ∗ = ϕ x ∗ .
b k
a
i=1 ui
(3.13)
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G. J. ZALMAI
Since x was an arbitrary feasible solution, we conclude from the above inequality that x ∗ is an optimal solution of (P).
An examination of the above proof suggests the following modification of
Theorem 3.4. Its proof is almost identical to that of Theorem 3.4.
Theorem 3.5. Consider the assumptions in Theorem 3.4 except that (3.3) is
replaced by either one of the following inequalities:
∗
∗ T
u∗
+ α∗i (t)T Ai (t)
i ∇2 fi t, x , ẋ
k
i=1
+
T
− λ∗ ∇2 gi t, x ∗ , ẋ ∗ − β∗i (t)T Bi (t)
T
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + γ ∗j (t)T Cj (t)
m
j=1
−D
k
u∗
i
T
T ∇3 fi t, x ∗ , ẋ ∗ − λ∗ ∇3 gi t, x ∗ , ẋ ∗
i=1
+
m
vj∗ (t)∇3 hj
T
t, x ∗ , ẋ ∗
(3.14)
x − x∗
j=1
≥0
b
k
a
∀t ∈ [a, b], ∀x ∈ F,
∗
∗ T
u∗
+ α∗i (t)T Ai (t)
i ∇2 fi t, x , ẋ
i=1
+
T
x − x∗
− λ∗ ∇2 gi t, x ∗ , ẋ ∗ − β∗i (t)T Bi (t)
m
T
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + γ ∗j (t)T Cj (t) x − x ∗
j=1
+
k
u∗
i
T
T ∇3 fi t, x ∗ , ẋ ∗ − λ∗ ∇3 gi t, x ∗ , ẋ ∗
i=1
+
m
vj∗ (t)∇3 hj
(3.15)
∗
∗ T
∗
ẋ − ẋ
t, x , ẋ
dt
j=1
≥ 0 ∀x ∈ F.
Then x ∗ is an optimal solution of (P).
Although Theorems 3.4 and 3.5 have almost identical proofs, it should be
stressed that (3.3), (3.14), and (3.15) are essentially different conditions. First, it
is evident that any (x, λ, u, v, α1 , . . . , αk , β1 , . . . , βk , γ 1 , . . . , γ m ) that satisfies the
conditions specified in Theorem 3.4 also satisfies the requirements of Theorem
3.5, but the converse is not necessarily true; second, (3.14) and (3.15) are not,
in general, transformable to (3.3); and third, (3.3) is a system of n equations,
whereas (3.14) and (3.15) are single inequalities. Evidently, from a computational point of view (3.3) is preferable to (3.14) and (3.15) because of the dependence of the latter two on the feasible set of (P).
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4155
The optimality conditions stated in Theorems 3.2, 3.4, and 3.5 contain the
parameter λ∗ which was introduced as a result of our indirect approach requiring an auxiliary parametric problem. However, reviewing the structure of
these optimality conditions, one can easily see that this parameter can, in fact,
be eliminated. Indeed, this can be accomplished simply by solving for λ∗ in
(3.5), substituting the result into (3.3), simplifying, and redefining the multiplier vector. This process leads to the following parameter-free versions of
Theorems 3.2, 3.4, and 3.5.
Theorem 3.6. A feasible solution x ◦ of (P) is optimal if and only if there
◦i
∈ PWSpi [a, b], β◦i ∈ PWSqi [a, b], i ∈ k,
exist u◦ ∈ U , v ◦ ∈ PWSm
+ [a, b], α
r
γ ◦j ∈ PWS j [a, b], j ∈ m, such that the following relations hold for all t ∈ [a, b]:
k
u◦i Ψ x ◦ , u◦ ∇2 fi t, x ◦ , ẋ ◦ + Ai (t)T α◦i (t)
i=1
− Φ x ◦ , u◦ ∇2 gi t, x ◦ , ẋ ◦ − Bi (t)T β◦i (t)
m
+
vj◦ (t) ∇2 hj t, x ◦ , ẋ ◦ + Cj (t)T γ ◦j (t)
j=1
−D
k
(3.16)
u◦i Ψ x ◦ , u◦ ∇3 fi t, x ◦ , ẋ ◦
i=1
− Φ x ◦ , u◦ ∇3 gi t, x ◦ , ẋ ◦
m
= 0,
vj◦ (t)∇3 hj t, x ◦ , ẋ ◦
+
j=1
m
j=1
◦i ∗
α (t)
L(i)
≤ 1,
vj◦ (t) hj t, x ◦ , ẋ ◦ + Cj (t)x ◦ N(j) = 0,
◦ Φ x ◦ , u◦
ϕ x = ◦ ◦ ,
Ψ x ,u
◦i ∗
◦j
β (t)
γ (t)∗ ≤ 1,
i ∈ k,
M(i) ≤ 1,
N(j)
α◦i (t)T Ai (t)x ◦ = Ai (t)x ◦ L(i) ,
β◦i (t)T Bi (t)x ◦ = Bi (t)x ◦ M(i) ,
γ ◦i (t)T Cj (t)x ◦ = Cj (t)x ◦ N(j) ,
j ∈ m,
(3.17)
(3.18)
j ∈ m,
(3.19)
i ∈ k,
(3.20)
(3.21)
where
Φ x ◦ , u◦ =
Ψ x ◦ , u◦ =
b k
a
i=1
b k
a
u◦i fi t, x ◦ , ẋ ◦ + Ai (t)x ◦ L(i) dt,
i=1
u◦i
gi t, x ◦ , ẋ ◦ − Bi (t)x ◦ M(i) dt.
(3.22)
4156
G. J. ZALMAI
Theorem 3.7. A feasible solution x ◦ of (P) is optimal if and only if there
◦i
∈ PWSpi [a, b], β◦i ∈ PWSqi [a, b], i ∈ k,
exist u◦ ∈ U , v ◦ ∈ PWSm
+ [a, b], α
rj
◦j
γ ∈ PWS [a, b], j ∈ m, such that (3.16), (3.17), (3.18), (3.19), (3.20), (3.21),
and either of the following inequalities hold for all t ∈ [a, b]:
T
u◦i Ψ x ◦ , u◦ ∇2 fi t, x ◦ , ẋ ◦ + α◦i (t)T Ai (t)
k
i=1
T
− Φ x ◦ , u◦ ∇2 gi t, x ◦ , ẋ ◦ − β◦i (t)T Bi (t)
+
m
T
vj◦ (t) ∇2 hj t, x ◦ , ẋ ◦ + γ ◦j (t)T Cj (t)
j=1
−D
k
T
u◦i Ψ x ◦ , u◦ ∇3 fi t, x ◦ , ẋ ◦
i=1
T − Φ x ◦ , u◦ ∇3 gi t, x ◦ , ẋ ◦
m
◦
◦
◦ T
x − x◦
vj (t)∇3 hj t, x , ẋ
+
j=1
≥0
∀x ∈ F,
b
k
a
T
u◦i Ψ x ◦ , u◦ ∇2 fi t, x ◦ , ẋ ◦ + α◦i (t)T Ai (t)
i=1
+
(3.23)
T
x − x◦
− Φ x ◦ , u◦ ∇2 gi t, x ◦ , ẋ ◦ − β◦i (t)T Bi (t)
m
T
vj◦ (t) ∇2 hj t, x ◦ , ẋ ◦ + γ ◦j (t)T Cj (t) x − x ◦
j=1
+
k
T
T u◦i Ψ x ◦ , u◦ ∇3 fi t, x ◦ , ẋ ◦ − Φ x ◦ , u◦ ∇3 gi t, x ◦ , ẋ ◦
i=1
+
m
vj◦ (t)∇3 hj
◦
◦ T
◦
ẋ − ẋ
t, x , ẋ
dt
j=1
≥0
∀x ∈ F.
4. Duality model I. Making use of Theorems 3.2, 3.4, and 3.5, in this section
we formulate two parametric dual problems for (P) and prove weak, strong, and
strict converse duality theorems. A number of parameter-free dual problems
will be discussed in Sections 5 and 6.
Consider the following two problems:
(DI) Maximize λ subject to
y(a) = x a ,
y(b) = x b ,
(4.1)
4157
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
ui ∇2 fi t, y, ẏ + Ai (t)T αi (t) − λ ∇2 gi t, y, ẏ − Bi (t)T βi (t)
k
i=1
vj (t) ∇2 hj t, y, ẏ + Cj (t)T γ j (t)
m
+
j=1
−D
i=1
+
m
= 0,
vj (t)∇3 hj t, y, ẏ
t ∈ [a, b],
j=1
b
k
a
(4.2)
ui ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
k
ui fi t, y, ẏ + αi (t)T Ai (t)y − λ gi t, y, ẏ − βi (t)T Bi (t)y
i=1
+
m
j=1
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y dt ≥ 0,
(4.3)
i ∗
i ∗
α (t)
β (t)
t ∈ [a, b], i ∈ k,
L(i) ≤ 1,
M(i) ≤ 1,
∗
j
γ (t)
t ∈ [a, b], j ∈ m,
N(j) ≤ 1,
y ∈ PWSn [a, b],
αi ∈ PWSpi [a, b],
λ ∈ R+ ,
βi ∈ PWSqi [a, b],
v ∈ PWSm
+ [a, b],
u ∈ U,
i ∈ k,
(4.4)
γ j ∈ PWSrj [a, b],
j ∈ m;
(4.5)
(D̃I) Maximize λ subject to (4.1), (4.3), (4.4), (4.5), and
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t) − λ ∇2 gi t, y, ẏ − βi (t)T Bi (t)
k
i=1
+
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
j=1
−D
T
T ui ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
k
i=1
+
m
T
vj (t)∇3 hj t, y, ẏ
(x − y)
j=1
≥0
∀t ∈ [a, b], x ∈ F,
or
b
k
a
i=1
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
T
− λ ∇2 gi t, y, ẏ − βi (t)T Bi (t) (x − y)
(4.6)
4158
G. J. ZALMAI
+
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t) (x − y)
j=1
+
k
T
T ui ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
i=1
m
+
T ẋ − ẏ dt ≥ 0
vj (t)∇3 hj t, y, ẏ
∀x ∈ F.
j=1
(4.7)
Evidently, the structures of (DI) and (D̃I) are motivated primarily by the nature and contents of the optimality conditions established in Theorems 3.2,
3.4, and 3.5 which form the basis for the proofs of all the duality relations for
(P)-(DI) and (P)-(D̃I).
Comparing (DI) and (D̃I), we observe that (D̃I) is relatively more general than
(DI) in the sense that any feasible solution of (DI) is also feasible for (D̃I), but
the converse is not necessarily true. Moreover, we see that (4.2) is a system of n
equations, whereas (4.6) and (4.7) are two inequalities which in general cannot
be expressed as equivalent systems of equations. Evidently, (DI) is preferable
to (D̃I) from a computational point of view because of the dependence of the
latter on the feasible set of (P). However, despite these apparent differences, it
turns out that all the duality results that can be established for (P)-(DI) are also
valid for (P)-(D̃I). Therefore, in the sequel we will consider only the pair (P)-(DI).
For the sake of simplicity of notation, we will henceforth let α = (α1 , . . . , αr ),
β = (β1 , . . . , βr ), and γ = (γ 1 , . . . , γ r ).
The next two theorems show that (DI) is a dual problem for (P).
Theorem 4.1 (weak duality). Let x and (y, λ, u, v, α, β, γ) be arbitrary feasible solutions of (P) and (DI), respectively. Then ϕ(x) ≥ λ.
Proof. Keeping in mind that λ ≥ 0, u ≥ 0, and v(t) ≥ 0 for all t ∈ [a, b],
we have
b
k
a
ui fi t, x, ẋ + Ai (t)x L(i) − λ gi t, x, ẋ − Bi (t)x M(i)
i=1
−
k
ui fi t, y, ẏ + αi (t)T Ai (t)y
i=1
≥
b k
a
i=1
− λ gi t, y, ẏ − βi (t)T Bi (t)y
dt
T
T ui ∇2 fi t, y, ẏ (x − y) + ∇3 fi t, y, ẏ
ẋ − ẏ
T
T − λ ∇2 gi t, y, ẏ (x − y) + ∇3 gi t, y, ẏ
ẋ − ẏ
+ Ai (t)x L(i) + λBi (t)x M(i)
− αi (t)T Ai (t)y − λβi (t)T Bi (t)y dt
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
=
b
k
a
4159
by the convexity of fi (t, ·, ·) and −gi (t, ·, ·), i ∈ k
ui
i=1
−
T
T ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
ẋ − ẏ + Ai (t)x L(i)
+ λBi (t)x M(i) − αi (t)T Ai (t)y − λβi (t)T Bi (t)y
k
ui αi (t)T Ai (t) + λβi (t)T Bi (t)T
i=1
+
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
j=1
−D
k
T
T ui ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
i=1
m
T
vj (t)∇3 hj t, y, ẏ
+
≥
b
k
a
(x − y) dt
j=1
ui
i=1
−
T
T ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
ẋ − ẏ + Ai (t)x L(i)
+ λBi (t)x M(i) − αi (t)T Ai (t)y − λβi (t)T Bi (t)y
k
∗ ∗ ui αi (t)L(i) Ai (t)x L(i) + λβi (t)M(i) Bi (t)x M(i)
i=1
− αi (t)T Ai (t)y − λβi (t)T Bi (t)y
m
T
vj (t) ∇2 hj t, y, ẏ (x − y)
+
j=1
+
k
T
T ui ∇3 fi t, y, ẏ − λ∇3 gi t, y, ẏ
+
j=1
≥
≥
b m
a
j=1
b m
a
j=1
b m
a
∗ + γ j (t)N(j) Cj (t)x N(j) − γ j (t)T Cj (t)y
i=1
m
≥−
by (4.2)
T vj (t)∇3 hj t, y, ẏ
ẋ − ẏ
dt
by (3.9) and integration by parts
T
T vj (t) ∇2 hj t, y, ẏ (x − y) + ∇3 hj t, y, ẏ
ẋ − ẏ
+ Cj (t)x N(j) − γ j (t)T Cj (t)y dt
by (4.4)
vj (t) hj t, y, ẏ − hj t, x, ẋ − Cj (t)x N(j)
+ γ j (t)T Cj (t)y dt
by the convexity of hj (t, ·, ·), j ∈ m
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y dt
j=1
(by the primal feasibility of x).
(4.8)
4160
G. J. ZALMAI
In view of (4.3), the above inequality reduces to
b k
a
i=1
ui fi t, x, ẋ + Ai (t)x L(i)
− λ gi t, x, ẋ + Bi (t)x M(i) dt ≥ 0.
(4.9)
Now using this inequality and Lemma 3.3, we see, as in the proof of Theorem
3.4, that ϕ(x) ≥ λ.
Theorem 4.2 (strong duality). Let x ∗ be an optimal solution of (P). Then
∗i
∈ PWSpi [a, b], β∗i ∈
there exist λ∗ ∈ R+ , u∗ ∈ U , v ∗ ∈ PWSm
+ [a, b], α
qi
r
PWS [a, b], i ∈ k, γ ∗j ∈ PWS j [a, b], j ∈ m, such that z∗ ≡ (x ∗ , λ∗ , u∗ , v ∗ ,
α∗ , β∗ , γ ∗ ) is an optimal solution of (DI) and ϕ(x ∗ ) = λ∗ .
Proof. By Theorem 3.2, there exist λ∗ (= ϕ(x ∗ )), u∗ , v ∗ , α∗ , β∗ , and γ ∗ ,
as specified above, such that z∗ is a feasible solution of (DI). Since ϕ(x ∗ ) = λ∗ ,
optimality of z∗ for (DI) follows from Theorem 4.1.
We also have the following converse duality result for (P)-(DI).
Theorem 4.3 (strict converse duality). Let x ∗ and (x̃, λ̃, ũ, ṽ, α̃, β̃, γ̃) be
optimal solutions of (P) and (DI), respectively, and assume that fi (t, ·, ·) or
−gi (t, ·, ·) is strictly convex throughout [a, b] for at least one index i ∈ k with
the corresponding component ũi of ũ positive, or hj (t, ·, ·) is strictly convex
throughout [a, b] for at least one j ∈ m with the corresponding component
ṽj (t) of ṽ(t) positive on [a, b]. Then x̃(t) = x ∗ (t) for all t ∈ [a, b], that is, x̃
is an optimal solution of (P), and ϕ(x ∗ ) = λ̃.
Proof. Suppose to the contrary that x̃ = x ∗ on a subset of [a, b] with positive length. From Theorem 4.2 we know that there exist λ∗ ∈ R+ , u∗ ∈ U, v ∗ ∈
∗i
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b], i ∈ k, γ ∗j ∈ PWSrj [a, b],
PWSm
+ [a, b], α
j ∈ m, such that (x ∗ , λ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ) is an optimal solution of (DI) and
ϕ(x ∗ ) = λ∗ . Now proceeding as in the proof of Theorem 4.1 (with x replaced
by x ∗ and (x, λ, u, v, α, β, γ) by (x̃, λ̃, ũ, ṽ, α̃, β̃, γ̃)), we arrive at the strict inequality
fi t, x ∗ , ẋ ∗ + Ai (t)x ∗ L(i) dt
> λ̃.
b k
∗
∗ − B (t)x ∗ i
i=1 ũi gi t, x , ẋ
a
M(i) dt
b k
a
i=1 ũi
(4.10)
Using this inequality and Lemma 3.3, we find, as in the proof of Theorem 4.1,
that ϕ(x ∗ ) > λ̃, which contradicts the fact that ϕ(x ∗ ) = λ∗ = λ̃. Therefore, we
must have x̃(t) = x ∗ (t), for all t ∈ [a, b], and ϕ(x ∗ ) = λ̃.
5. Duality model II. In the remainder of this paper we will formulate and
discuss several parameter-free duality models for (P) whose forms and features
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4161
are based on Theorems 3.6 and 3.7. We begin with the following pair of dual
problems:
(DII)
fi t, y, ẏ + Ai (t)y L(i) dt
Maximize b k
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
a
b k
a
i=1 ui
(5.1)
subject to
y(a) = x a ,
k
Ψ (y, u)
y(b) = x b ,
(5.2)
ui ∇2 fi t, y, ẏ + Ai (t)T αi (t)
i=1
− Φ(y, u)
k
ui ∇2 gi t, y, ẏ − Bi (t)T βi (t)
i=1
+
m
vj (t) ∇2 hj t, y, ẏ + Cj (t)T γ j (t)
(5.3)
j=1
−D
k
ui Ψ (y, u)∇3 fi t, y, ẏ − Φ(y, u)∇3 gi t, y, ẏ
i=1
+
m
= 0,
vj (t)∇3 hj t, y, ẏ
t ∈ [a, b],
j=1
m
vj (t) hj t, y, ẏ + Cj (t)y N(j) ≥ 0,
t ∈ [a, b],
(5.4)
j=1
i ∗
i ∗
α (t)
β (t)
t ∈ [a, b], i ∈ k,
L(i) ≤ 1,
M(i) ≤ 1,
(5.5)
∗
j
γ (t)
t ∈ [a, b], j ∈ m,
N(j) ≤ 1,
αi (t)T Ai (t)y=Ai (t)y L(i) , βi (t)T Bi (t)y=Bi (t)y M(i) , t ∈ [a, b], i ∈ k,
(5.6)
(5.7)
γ j (t)T Cj (t)y = Cj (t)y N(j) , t ∈ [a, b], j ∈ m,
y ∈ PWSn [a, b],
u ∈ U,
βi ∈ PWSqi [a, b],
v ∈ PWSm
+ [a, b],
i ∈ k,
αi ∈ PWSpi [a, b],
γ j ∈ PWSrj [a, b],
j ∈ m,
(5.8)
where Φ and Ψ are as defined in Theorem 3.6;
(D̃II)
fi t, y, ẏ + Ai (t)y L(i) dt
Maximize b k
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
a
b k
a
i=1 ui
(5.9)
4162
G. J. ZALMAI
subject to (5.2), (5.4), (5.5), (5.6), (5.7), (5.8), and
k
Ψ (y, u)
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
i=1
k
− Φ(y, u)
T
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t)
i=1
+
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
(5.10)
j=1
−D
T
T ui Ψ (y, u)∇3 fi t, y, ẏ − Φ(y, u)∇3 gi t, y, ẏ
k
i=1
+
m
vj (t)∇3 hj t, y, ẏ
T
(x − y) ≥ 0
∀t ∈ [a, b], x ∈ F,
j=1
or
b k
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
Ψ (y, u)
a
i=1
− Φ(y, u)
k
ui ∇2 gi t, y, ẏ
T
− β (t) Bi (t) (x − y)
i
T
i=1
+
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t) (x − y)
(5.11)
j=1
+
k
T
T ui Ψ (y, u)∇3 fi t, y, ẏ − Φ(y, u)∇3 gi t, y, ẏ
i=1
+
m
T ẋ − ẏ dt ≥ 0
vj (t)∇3 hj t, y, ẏ
∀x ∈ F.
j=1
The remarks made earlier about the relationships between (DI) and (D̃I) are,
of course, also applicable to (DII) and (D̃II).
Throughout this section and the next one, it will be assumed that Φ(y, u) ≥ 0
and Ψ (y, u) > 0, for all (y, u) such that (y, u, v, α, β, γ) is a feasible solution
of the dual problem under consideration.
We next proceed to state and prove weak, strong, and strict converse duality
theorems for (P)-(DII).
Theorem 5.1 (weak duality). Let x and z ≡ (y, u, v, α, β, γ) be arbitrary
feasible solutions of (P) and (DII), respectively. Then ϕ(x) ≥ ψ(z), where ψ is
the objective function of (DII).
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4163
Proof. Keeping in mind that u ≥ 0, Φ(y, u) ≥ 0, Ψ (y, u) > 0, and v(t) ≥ 0
for all t ∈ [a, b], we have
b k
a
k
b ui gi t, y, ẏ − Bi (t)y M(i) dt
ui fi t, x, ẋ + Ai (t)x L(i) dt
a
i=1
−
×
b k
a
ui fi t, y, ẏ + Ai (t)y L(i) dt
i=1
b k
a
i=1
ui gi t, x, ẋ − Bi (t)x M(i) dt
i=1
= Ψ (y, u)
k
b
a
−
i=1
b k
a
− Φ(y, u)
a
b k
a
− Φ(y, u)
ui fi t, y, ẏ + Ai (t)y L(i) dt
ui gi t, x, ẋ − Bi (t)x M(i) dt
i=1
b k
a
ui gi t, y, ẏ − Bi (t)y M(i) dt
i=1
T
T ui ∇2 fi t, y, ẏ (x − y) + ∇3 fi t, y, ẏ
ẋ − ẏ
i=1
+ Ai (t)x L(i) − Ai (t)y L(i) dt
b k
a
i=1
k
b
−
≥ Ψ (y, u)
ui fi t, x, ẋ + Ai (t)x L(i) dt
T
T ui ∇2 gi t, y, ẏ (x − y) + ∇3 gi t, y, ẏ
ẋ − ẏ
i=1
− Bi (t)x M(i) + Bi (t)y M(i) dt
by the convexity of fi (t, ·, ·) and −gi (t, ·, ·), i ∈ k
b
=
a
Ψ (y, u)
k
T ẋ − ẏ − αi (t)T Ai (t)(x − y)
ui ∇3 fi t, y, ẏ
i=1
− Φ(y, u)
k
i=1
−
m
j=1
+ Ai (t)x L(i) − Ai (t)y L(i)
T ẋ − ẏ + βi (t)T Bi (t)(x − y)
ui ∇3 gi t, y, ẏ
− Bi (t)x M(i) + Bi (t)y M(i)
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t) (x − y)
4164
G. J. ZALMAI
k
+ D Ψ (y, u)
T
ui ∇3 fi t, y, ẏ
i=1
− Φ(y, u)
k
T
ui ∇3 gi t, y, ẏ
i=1
+
T
(x − y) dt
vj (t)∇3 hj t, y, ẏ
m
by (5.3)
j=1
b
≥
Ψ (y, u)
a
∗
T ui ∇3 fi t, y, ẏ
ẋ − ẏ − Ai (t)x L(i) αi (t)L(i)
k
i=1
− Φ(y, u)
+ αi (t)T Ai (t)y + Ai (t)x L(i) − Ai (t)y L(i)
k
i=1
−
m
∗
T ui ∇3 gi t, y, ẏ
ẋ − ẏ + Bi (t)x M(i) βi (t)M(i)
− βi (t)T Bi (t)y − Bi (t)x M(i) + Bi (t)y M(i)
∗
T vj (t) ∇2 hj t, y, ẏ + Cj (t)x N(j) γ j (t)N(j)
j=1
− γ j (t)T Cj (t)y
−
k
T
T ui Ψ (y, u)∇3 fi t, y, ẏ − Φ(y, u)∇3 gi t, y, ẏ
i=1
+
T vj (t)∇3 hj t, y, ẏ
ẋ − ẏ dt
m
j=1
by (3.9) and integration by parts
≥−
b m
a
T
T vj (t) ∇2 hj t, y, ẏ (x − y) + ∇3 hj t, y, ẏ
ẋ − ẏ
j=1
+ Cj (t)x N(j) − γ j (t)T Cj (t)y dt
by (5.5) and (5.6)
≥
b m
a
≥0
vj (t) hj t, y, ẏ −hj t, x, ẋ −Cj (t)x N(j) +γ j (t)T Cj (t)y dt
j=1
by the convexity of hj (t, ·, ·), j ∈ m
by the primal feasibility of x, (5.4), and (5.7) .
(5.12)
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4165
Now it follows from Lemma 3.3 and the above inequality that
fi t, x, ẋ + Ai (t)x L(i) dt
ϕ(x) = max b k
c∈U
i=1 ci gi t, x, ẋ − Bi (t)x M(i) dt
a
b k
i=1 ui fi t, x, ẋ + Ai (t)x L(i) dt
a
≥ b k
i=1 ui gi t, x, ẋ − Bi (t)x M(i) dt
a
b k
i=1 ui fi t, y, ẏ + Ai (t)y L(i) dt
a
≥ b k
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
a
b k
a
i=1 ci
(5.13)
= ψ(y, u, v, α, β, γ).
Theorem 5.2 (strong duality). Let x ∗ be an optimal solution of (P). Then
∗i
there exist u∗ ∈ U , v ∗ ∈ PWSm
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b],
+ [a, b], α
r
i ∈ k, γ ∗j ∈ PWS j [a, b], j ∈ m, such that z∗ ≡ (x ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ) is an
optimal solution of (DII) and ϕ(x ∗ ) = ψ(z∗ ).
Proof. By Theorem 3.6, there exist u∗ , v ∗ , α∗ , β∗ , and γ ∗ , as specified
above, such that z∗ is a feasible solution of (DII). Since ϕ(x ∗ ) = Φ(x ∗ ,
u∗ )/Ψ (x ∗ , u∗ ) = ψ(z∗ ), optimality of z∗ for (DII) follows from Theorem 5.1.
Theorem 5.3 (strict converse duality). Let x ∗ and z̃ ≡ (x̃, ũ, ṽ, α̃, β̃, γ̃) be
optimal solutions of (P) and (DII), respectively, and assume that fi (t, ·, ·) or
−gi (t, ·, ·) is strictly convex throughout [a, b] for at least one index i ∈ k with the
corresponding component ũi of ũ (and Φ(x̃, ũ)) positive, or hj (t, ·, ·) is strictly
convex throughout [a, b] for at least one j ∈ m with the corresponding component ṽj (t) of ṽ(t) positive on [a, b]. Then x̃(t) = x ∗ (t) for all t ∈ [a, b], that
is, x̃ is an optimal solution of (P) and ϕ(x ∗ ) = ψ(z̃).
Proof. Suppose to the contrary that x̃(t) = x ∗ (t) on a subset of [a, b]
with positive length. From Theorem 5.2 we know that there exist u∗ ∈ U, v ∗ ∈
∗i
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b], i ∈ k, γ ∗j ∈ PWSrj [a, b],
PWSm
+ [a, b], α
j ∈ m, such that z∗ ≡ (x ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ) is an optimal solution of (DII)
and ϕ(x ∗ ) = ψ(z∗ ). Now proceeding as in the proof of Theorem 5.1 (with x
replaced by x ∗ and z by z̃), we arrive at the strict inequality
Φ x̃, ũ
Φ x ∗ , ũ
> .
Ψ x ∗ , ũ
Ψ x̃, ũ
(5.14)
Using this inequality and Lemma 3.3, it can be shown, as in the proof of
Theorem 5.1, that ϕ(x ∗ ) > ψ(z̃), in contradiction to the fact that ϕ(x ∗ ) =
ψ(z∗ ) = ψ(z̃). Therefore, it follows that x̃(t) = x ∗ (t) for all t ∈ [a, b] and
ϕ(x ∗ ) = ψ(z̃).
4166
G. J. ZALMAI
Next, we turn to a brief discussion of certain variants of (DII) and (D̃II). We
show that the constraints (5.6) and (5.7) are superfluous and can be deleted
without invalidating the foregoing duality results. More precisely, we demonstrate that the following reduced versions of (DII) and (D̃II) are also dual problems for (P):
(DIIA)
b k
ui fi t, y, ẏ + αi (t)T Ai (t)y dt
i
T
i=1 ui gi t, y, ẏ − β (t) Bi (t)y dt
Maximize ab i=1
k
a
(5.15)
subject to
y(a) = x a ,
k
y(b) = x b ,
(5.16)
ui Γ (y, u, β) ∇2 fi t, y, ẏ + Ai (t)T αi (t)
i=1
− Π(y, u, α) ∇2 gi t, y, ẏ − Bi (t)T βi (t)
+
m
vj (t) ∇2 hj t, y, ẏ + Cj (t)T γ j (t)
j=1
−D
k
(5.17)
ui Γ (y, u, β)∇3 fi t, y, ẏ
i=1
− Π(y, u, α)∇3 gi t, y, ẏ
m
+
vj (t)∇3 hj t, y, ẏ
= 0, t ∈ [a, b],
j=1
m
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y ≥ 0,
t ∈ [a, b],
(5.18)
j=1
i ∗
i ∗
α (t)
β (t)
t ∈ [a, b], i ∈ k,
L(i) ≤ 1,
M(i) ≤ 1,
∗
j
γ (t)
t ∈ [a, b], j ∈ m,
N(j) ≤ 1,
y ∈ PWSn [a, b],
v ∈ PWSm
+ [a, b],
u ∈ U,
βi ∈ PWSqi [a, b],
i ∈ k,
αi ∈ PWSpi [a, b],
γ j ∈ PWSrj [a, b],
j ∈ m,
(5.19)
(5.20)
where
Π(y, u, α) =
Γ (y, u, β) =
b k
a
i=1
b k
a
ui fi t, y, ẏ + αi (t)T Ai (t)y dt,
i=1
ui gi t, y, ẏ − βi (t)T Bi (t)y dt;
(5.21)
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4167
(D̃IIA)
b k
ui fi t, y, ẏ + αi (t)T Ai (t)y dt
i
T
i=1 ui gi t, y, ẏ − β (t) Bi (t)y dt
Maximize ab i=1
k
a
(5.22)
subject to (5.16), (5.18), (5.19), (5.20), and
T
ui Γ (y, u, β) ∇2 fi t, y, ẏ + αi (t)T Ai (t)
k
i=1
T
− Π(y, u, α) ∇2 gi t, y, ẏ − βi (t)T Bi (t)
+
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
m
(5.23)
j=1
−D
T
T ui Γ (y, u, β)∇3 fi t, y, ẏ − Π(y, u, α)∇3 gi t, y, ẏ
k
i=1
m
+
vj (t)∇3 hj t, y, ẏ
T
(x − y) ≥ 0
∀t ∈ [a, b], x ∈ F,
j=1
or
b k
a
T
ui Γ (y, u, β) ∇2 fi t, y, ẏ + αi (t)T Ai (t)
i=1
T
− Π(y, u, α) ∇2 gi t, y, ẏ − βi (t)T Bi (t)
m
T
j
T
vj (t) ∇2 hj t, y, ẏ + γ (t) Cj (t) (x − y)
+
j=1
+
T
T ui Γ (y, u, β)∇3 fi t, y, ẏ − Π(y, u, α)∇3 gi t, y, ẏ
k
i=1
+
(5.24)
m
T ẋ − ẏ dt ≥ 0
vj (t)∇3 hj t, y, ẏ
∀x ∈ F.
j=1
Since it may not be immediately apparent that (DIIA) and (D̃IIA) are dual
problems for (P), we provide a proof for (P)-(DIIA).
Throughout this section and the next one, we assume that Π(y, u, α) ≥ 0
and Γ (y, u, β) > 0, for all y, u, α, and β such that (y, u, α, β, γ) is a feasible
solution of the dual problem under consideration.
Theorem 5.4 (weak duality). Let x and (y, u, v, α, β, γ) be arbitrary feasible
solutions of (P) and (DIIA), respectively. Then ϕ(x) ≥ ω(y, u, v, α, β, γ), where
ω is the objective function of (DIIA).
4168
G. J. ZALMAI
Proof. Keeping in mind that u ≥ 0, Π(y, u, α) ≥ 0, Γ (y, u, β) > 0, and
v(t) ≥ 0 for all t ∈ [a, b], we have
b k
a
ui gi t, y, ẏ − βi (t)T Bi (t)y dt
i=1
−
×
b k
a
ui gi t, x, ẋ − Bi (t)x M(i) dt
i=1
= Γ (u, y, β)
k
b
a
−
i=1
a
− Π(y, u, α)
=
i=1
i=1
ui gi t, x, ẋ − Bi (t)x M(i) dt
i=1
b k
a
ui gi t, y, ẏ − βi (t)T Bi (t)y dt
i=1
+ Ai (t)x L(i) − αi (t)Ai (t)y
T
T ẋ − ẏ
− Π(y, u, α) ∇2 gi t, y, ẏ (x − y) + ∇3 gi t, y, ẏ
− Bi (t)x M(i) + βi (t)T Bi (t)y dt
by the convexity of fi (t, ·, ·) and −gi (t, ·, ·), i ∈ k
T ẋ − ẏ − αi (t)T Ai (t)(x − y)
ui Γ (y, u, β) ∇3 fi t, y, ẏ
i=1
−
T
T ui Γ (y, u, β) ∇2 fi t, y, ẏ (x − y) + ∇3 fi t, y, ẏ
ẋ − ẏ
b
k
a
ui fi t, y, ẏ + αi (t)T Ai (t)y dt
k
b
a
−
b k
ui fi t, x, ẋ + Ai (t)x L(i) dt
b k
a
i=1
ui fi t, y, ẏ + αi (t)T Ai (t)y dt
≥
a
ui fi t, x, ẋ + Ai (t)x L(i) dt
i=1
b k
a
b k
+ Ai (t)x L(i) − αi (t)T Ai (t)y
T ẋ − ẏ + βi (t)T Bi (t)(x − y)
− Π(y, u, α) ∇3 gi t, y, ẏ
− Bi (t)x M(i) + βi (t)T Bi (t)y
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
j=1
−D
k
i=1
T
T ui Γ (y, u, β)∇3 fi t, y, ẏ − Π(y, u, α)∇3 gi t, y, ẏ
4169
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
+
m
T
vj (t)∇3 hj t, y, ẏ
(x − y) dt
by (5.17)
j=1
≥
b
k
a
T ui Γ (y, u, β) ∇3 fi t, y, ẏ
ẋ − ẏ + Ai (t)x L(i)
i=1
∗ − Ai (t)x L(i) αi (t)L(i)
T − Π(y, u, α) ∇3 gi t, y, ẏ
ẋ − ẏ − Bi (t)x M(i)
∗ + Bi (t)x M(i) βi (t)M(i)
−
T
vj (t) ∇2 hj t, y, ẏ (x − y)
m
j=1
−
k
∗
+ Cj (t)x N(j) γ j (t)N(j) − γ j (t)T Cj (t)y
T
ui Γ (y, u, β)∇3 fi t, y, ẏ
i=1
T − Π(y, u, α)∇3 gi t, y, ẏ
m
T vj (t)∇3 hj t, y, ẏ
+
ẋ − ẏ dt
j=1
≥−
≥
b m
a
j=1
b m
a
by (3.9) and integration by parts
T
T ẋ − ẏ
vj (t) ∇2 hj t, y, ẏ (x − y) + ∇3 hj t, y, ẏ
+ Cj (t)x N(j) − γ j (t)T Cj (t)y dt
by (5.19)
vj (t) hj t, y, ẏ − hj t, x, ẋ
j=1
− Cj (t)x N(j) + γ j (t)T Cj (t)y dt
by the convexity of hj (t, ·, ·), j ∈ m
≥ 0 by the primal feasibility of x and (5.18) .
(5.25)
Hence
Π(y, u, α)
Φ(x, u)
≥
.
Ψ (x, u)
Γ (y, u, β)
(5.26)
Now using this inequality and Lemma 3.3, as in the proof of Theorem 5.1, we
obtain the desired inequality ϕ(x) ≥ ω(y, u, v, α, β, γ).
4170
G. J. ZALMAI
Theorem 5.5 (strong duality). Let x ∗ be an optimal solution of (P). Then
∗i
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b],
there exist u∗ ∈ U , v ∗ ∈ PWSm
+ [a, b], α
rj
∗j
i ∈ k, γ ∈ PWS [a, b], j ∈ m, such that (x ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ) is an optimal
solution of (DIIA) and ϕ(x ∗ ) = ω(x ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ).
Proof. The proof is similar to that of Theorem 5.2.
Theorem 5.6 (strict converse duality). Let x ∗ and (x̃, ũ, ṽ, α̃, β̃, γ̃) be optimal solutions of (P) and (DIIA), respectively, and assume that fi (t, ·, ·) or −gi (t, ·,
·) is strictly convex throughout [a, b] for at least one index i ∈ k with the corresponding component ũi of ũ (and Φ(x̃, ũ)) positive, or hj (t, ·, ·) is strictly
convex throughout [a, b] for at least one j ∈ m with the corresponding component ṽj (t) of ṽ(t) positive on [a, b]. Then x̃(t) = x ∗ (t) for all t ∈ [a, b], that
is, x̃ is an optimal solution of (P) and ϕ(x ∗ ) = ω(x̃, ũ, ṽ, α̃, β̃, γ̃).
Proof. The proof is similar to that of Theorem 5.3.
6. Duality model III. In this section, we show that the following variants of
(DII) are also dual problems for (P):
(DIII)
1
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
Maximize b k
a
×
b
k
a
ui fi t, y, ẏ + Ai (t)y L(i)
i=1
+
m
(6.1)
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
j=1
subject to
y(a) = x a ,
k
Ψ (y, u)
(6.2)
ui ∇2 fi t, y, ẏ + Ai (t)T αi (t)
i=1
+
y(b) = x b ,
m
T j
vj (t) ∇2 hj t, y, ẏ + Cj (t) γ (t)
j=1
k
− Φ(y, u) + Ω(y, v)
ui ∇2 gi t, y, ẏ − Bi (t)T βi (t)
− D Ψ (y, u)
i=1
k
m
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
j=1
k
− Φ(y, u) + Ω(y, v)
ui ∇3 gi t, y, ẏ
= 0,
i=1
t ∈ [a, b],
(6.3)
4171
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
m
vj (t) hj t, y, ẏ + Cj (t)y N(j) ≥ 0,
t ∈ [a, b],
(6.4)
j=1
i ∗
i ∗
α (t)
β (t)
t ∈ [a, b], i ∈ k,
L(i) ≤ 1,
M(i) ≤ 1,
∗
j
γ (t)
t ∈ [a, b], j ∈ m,
N(j) ≤ 1,
αi (t)T Ai (t)y=Ai (t)y L(i) ,
βi (t)T Bi (t)y=Bi (t)y M(i) ,
γ j (t)T Cj (t)y = Cj (t)y N(j) ,
y ∈ PWSn [a, b],
βi ∈ PWSqi [a, b],
(6.7)
αi ∈ PWSpi [a, b],
γ j ∈ PWSrj [a, b],
i ∈ k,
t ∈ [a, b], i ∈ k,
(6.6)
t ∈ [a, b], j ∈ m,
v ∈ PWSm
+ [a, b],
u ∈ U,
(6.5)
j ∈ m,
(6.8)
where Φ and Ψ are as defined in Theorem 3.6 and
Ω(y, v) =
b m
a
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt;
(6.9)
j=1
(D̃III)
1
Maximize b k
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
a
×
b
k
a
ui fi t, y, ẏ + Ai (t)y L(i)
i=1
+
m
(6.10)
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
j=1
subject to (6.2), (6.4), (6.5), (6.6), (6.7), (6.8), and
Ψ (y, u)
k
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
i=1
+
m
T
j
T
vj (t) ∇2 hj t, y, ẏ + γ (t) Cj (t)
j=1
k
T
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t)
− Φ(y, u) + Ω(y, v)
− D Ψ (y, u)
i=1
k
T
ui ∇3 fi t, y, ẏ +
i=1
m
T
vj (t)∇3 hj t, y, ẏ
j=1
k
T
− Φ(y, u) + Ω(y, v)
ui ∇3 gi t, y, ẏ
i=1
≥0
∀t ∈ [a, b], x ∈ F,
(x − y)
(6.11)
4172
G. J. ZALMAI
or
b k
Ψ (y, u)
a
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
i=1
+
m
T
j
T
vj (t) ∇2 hj t, y, ẏ + γ (t) Cj (t)
j=1
k
T
− Φ(y, u) + Ω(y, v)
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t) (x − y)
+ Ψ (y, u)
i=1
k
m
T T
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
j=1
k
T ẋ − ẏ dt
ui ∇3 gi t, y, ẏ
− Φ(y, u) + Ω(y, v)
i=1
≥ 0 ∀x ∈ F.
(6.12)
The remarks made about the relationships between (DI) and (D̃I) are, of
course, also applicable to (DIII) and (D̃III).
We now proceed to establish weak, strong, and strict converse duality relations for (P)-(DIII).
Theorem 6.1 (weak duality). Let x and (y, u, v, α, β, γ) be arbitrary feasible
solutions of (P) and (DIII), respectively. Then ϕ(x) ≥ ξ(y, u, v, α, β, γ), where ξ
is the objective function of (DIII).
Proof. Keeping in mind that u ≥ 0, Φ(y, u) ≥ 0, Ψ (y, u) > 0, and v(t) ≥ 0
for all t ∈ [a, b], we have
b k
a
k
b ui gi t, y, ẏ − Bi (t)y M(i) dt
ui fi t, x, ẋ + Ai (t)x L(i) dt
a
i=1
−
k
b
a
+
i=1
ui fi t, y, ẏ + Ai (t)y L(i)
i=1
m
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
j=1
×
b k
a
ui gi t, x, ẋ − Bi (t)x M(i) dt
i=1
= Ψ (y, u)
k
b
a
i=1
ui fi t, x, ẋ + Ai (t)x L(i)
− fi t, y, ẏ − Ai (t)y L(i) dt
4173
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
b k
− Φ(y, u)
a
ui gi t, x, ẋ − Bi (t)x M(i)
i=1
− gi t, y, ẏ + Bi (t)y M(i) dt
b m
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
− Ψ (x, u)
a
≥ Ψ (y, u)
b k
a
− Φ(y, u)
− Ψ (x, u)
T
T ẋ − ẏ
ui ∇2 fi t, y, ẏ (x − y) + ∇3 fi t, y, ẏ
i=1
+ Ai (t)x L(i) − Ai (t)y L(i) dt
b k
a
T
T ẋ − ẏ
ui ∇2 gi t, y, ẏ (x − y) + ∇3 gi t, y, ẏ
i=1
b m
a
j=1
− Bi (t)x M(i) + Bi (t)y M(i) dt
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
j=1
by the convexity of fi (t, ·, ·) and −gi (t, ·, ·), i ∈ k
k
b
T =
ui ∇3 fi t, y, ẏ
Ψ (y, u)
ẋ − ẏ − αi (t)T Ai (t)(x − y)
a
i=1
−
+ Ai (t)x L(i) − Ai (t)y L(i)
m
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t) (x − y)
j=1
T ui ∇3 gi t, y, ẏ
ẋ − ẏ + βi (t)T Bi (t)(x − y)
k
− Φ(y, u)
i=1
+ Ω(y, v)
k
i=1
− Bi (t)x M(i) + Bi (t)y M(i)
T
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t) (x − y)
+ D Ψ (y, u)
k
m
T T
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
− Φ(y, u) + Ω(y, v)
j=1
k
T
ui ∇3 gi t, y, ẏ
(x − y)
i=1
− Ψ (x, u)
b
≥
a
Ψ (y, u)
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
m
by (6.3)
j=1
k
i=1
∗
T ui ∇3 fi t, y, ẏ
ẋ − ẏ − Ai (t)x L(i) αi (t)L(i)
+ αi (t)T Ai (t)y + Ai (t)x L(i) − Ai (t)y L(i)
4174
G. J. ZALMAI
−
T
vj (t) ∇2 hj t, y, ẏ (x − y)
m
j=1
∗
T ẋ − ẏ + Bi (t)x M(i) βi (t)M(i)
ui ∇3 gi t, y, ẏ
k
− Φ(y, u)
∗
+ Cj (t)x N(j) γ j (t)N(j)
− γ j (t)T Cj (t)y
i=1
+ Ω(y, v)
− βi (t)T Bi (t)y − Bi (t)x M(i) + Bi (t)y M(i)
T
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t) (x − y)
k
i=1
k
− Ψ (y, u)
m
T T
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
j=1
− Φ(y, u) + Ω(y, v)
k
T ui ∇3 gi t, y, ẏ
ẋ − ẏ
i=1
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
m
− Ψ (x, u)
j=1
b
≥
− Ψ (y, u)
a
by (3.9) and integration by parts
m
j=1
+ Ω(y, v)
k
T
T vj (t) ∇2 hj t, y, ẏ (x − y) + ∇3 hj t, y, ẏ
ẋ − ẏ
+ Cj (t)x N(j) − γ j (t)T Cj (t)y
T
T ui ∇2 gi t, y, ẏ (x − y) + ∇3 gi t, y, ẏ
ẋ − ẏ
i=1
− βi (t)T Bi (t)(x − y)
m
− Ψ (x, u)
j=1
b
≥
a
Ψ (y, u)
m
by (6.5), (6.6), and (6.7)
vj (t) hj t, y, ẏ − hj t, x, ẋ
j=1
+ Ω(y, v)
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt
k
i=1
− Cj (t)x N(j) + γ j (t)T Cj (t)y
T
ui ∇2 gi t, y, ẏ (x − y)
T + ∇3 gi t, y, ẏ
ẋ − ẏ − βi (t)T Bi (t)(x − y)
4175
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
− Ψ (x, u)
m
vj (t) hj t, y, ẏ + Cj (t)y N(j)
j=1
dt
by the convexity of hj (t, ·, ·), j ∈ m
≥ Ω(y, v)Ψ (y, u) + Ω(y, v)
×
b k
a
T
ui − gi t, x, ẋ + ∇2 gi t, y, ẏ (x − y)
i=1
T + ∇3 gi t, y, ẏ
ẋ − ẏ + Bi (t)x M(i)
∗
− Bi (t)x M(i) βi (t)M(i) − βi (t)T Bi (t)y dt
by the primal feasibility of x,
definitions of Ψ (x, u) and Ω(y, v), (6.7), and (3.9)
≥ Ω(y, v)Ψ (y, u) + Ω(y, v)
b k
a
ui − gi t, y, ẏ + βi (t)T Bi (t)y dt
i=1
by the convexity of −gi (t, ·, ·), i ∈ k, (6.4), and (6.5)
= Ω(y, v)Ψ (y, u) − Ω(y, v)Ψ (y, u)
by the definition of Ψ (y, u) and (6.7)
= 0,
(6.13)
which leads to the inequality
fi t, x, ẋ + Ai (t)x L(i) dt
b k
i=1 ui gi t, x, ẋ − Bi (t)x M(i) dt
a
1
≥ b k
i=1 ui gi t, y, ẏ − Bi (t)y M(i) dt
a
b
k
×
ui fi t, y, ẏ + Ai (t)y L(i)
b k
i=1 ui
a
a
i=1
m
+
(6.14)
vj (t) hj t, y, ẏ + Cj (t)y N(j) dt.
j=1
Now invoking Lemma 3.3 and proceeding as in the proof of Theorem 5.1, we
obtain the desired inequality ϕ(x) ≥ ξ(y, u, v, α, β, γ).
Theorem 6.2 (strong duality). Let x ∗ be an optimal solution of (P). Then
∗i
∈ PWSpi [a, b], β∗i ∈ PWSqi [a, b],
there exist u∗ ∈ U , v ∗ ∈ PWSm
+ [a, b], α
rj
∗j
i ∈ k, γ ∈ PWS [a, b], j ∈ m, such that z∗ ≡ (x ∗ , u∗ , v ∗ , α∗ , β∗ , γ ∗ ) is an
optimal solution of (DIII) and ϕ(x ∗ ) = ξ(z∗ ).
4176
G. J. ZALMAI
Proof. Since x ∗ is an optimal solution of (P), by Theorem 3.6, there exist
u , α∗ , β∗ , and γ ∗ , as specified above, and v ◦ ∈ PWSm
+ [a, b] such that the
following relations hold for all t ∈ [a, b]:
∗
k
∗
∗
∇2 fi t, x ∗ , ẋ ∗ + Ai (t)T α∗i (t)
u∗
i Ψ x ,u
i=1
− Φ x ∗ , u∗ ∇2 gi t, x ∗ , ẋ ∗ − Bi (t)T β∗i (t)
+
m
vj◦ (t) ∇2 hj t, x ∗ , ẋ ∗ + Cj (t)T γ ∗j (t)
j=1
−D
k
∗ ∗
u∗
∇3 fi t, x ∗ , ẋ ∗ − Φ x ∗ , u∗ ∇3 gi t, x ∗ , ẋ ∗
i Ψ x ,u
i=1
+
m
vj◦ (t)∇3 hj
∗
∗
t, x , ẋ
= 0,
j=1
(6.15)
m
vj◦ (t) hj t, x ∗ , ẋ ∗ + Cj (t)x ∗ N(j) = 0,
(6.16)
j=1
∗i ∗
α (t)
L(i)
≤ 1,
Φ x ∗ , u∗
ϕ x∗ = ∗ ∗ ,
Ψ x ,u
∗i ∗
∗j
β (t)
γ (t)∗ ≤ 1,
i ∈ k,
M(i) ≤ 1,
N(j)
α∗i (t)T Ai (t)x ∗ = Ai (t)x ∗ L(i) ,
β∗i (t)T Bi (t)x ∗ = Bi (t)x ∗ M(i) ,
γ ∗j (t)T Cj (t)x ∗ = Cj (t)x ∗ N(j) ,
j ∈ m.
(6.17)
j ∈ m,
(6.18)
i ∈ k,
(6.19)
(6.20)
Since Ψ (x ∗ , u∗ ) > 0 and (6.16) holds, (6.15) and (6.16) can be rewritten as
follows:
k
∗
∗
u∗
Ψ x ∗ , u∗
+ Ai (t)T α∗i (t)
i ∇2 fi t, x , ẋ
i=1
vj◦ (t)
∗
∗
T
∗j
∇2 hj t, x , ẋ + Cj (t) γ (t)
+
Ψ x ∗ , u∗
j=1
∗ ∗
v◦
∗
− Φ x , u + Ω x , ∗ ∗
Ψ x ,u
m
×
k
i=1
∗
∗
u∗
− Bi (t)T β∗i (t)
i ∇2 gi t, x , ẋ
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
− D Ψ x∗, u
k
∗
4177
∗
∗
u∗
i ∇3 fi t, x , ẋ
i=1
vj◦ (t)
∗
∗
∇3 hj t, x , ẋ
+
Ψ x ∗ , u∗
j=1
k
v◦
∗
∗
− Φ x ∗ , u∗ + Ω x ∗ , ∗ ∗ u∗
∇
g
,
ẋ
t,
x
3 i
i
Ψ x ,u
i=1
m
= 0,
m
j=1
t ∈ [a, b],
vj◦ (t)
hj t, x ∗ , ẋ ∗ + Cj (t)x ∗ N(j)
Ψ x ∗ , u∗
(6.21)
= 0,
t ∈ [a, b].
(6.22)
Now letting v ∗ = v ◦ /Ψ (x ∗ , u∗ ) in (6.21) and (6.22), we see from (6.18), (6.19),
(6.20), (6.21), and (6.22) that z∗ is a feasible solution of (DIII). But in view
of (6.17) and (6.22), ϕ(x ∗ ) = ξ(z∗ ) and, therefore, by Theorem 6.1, z∗ is an
optimal solution of (DIII).
Theorem 6.3 (strict converse duality). Let x ∗ and z̃ ≡ (x̃, ũ, ṽ, α̃, β̃, γ̃) be
optimal solutions of (P) and (DIII), respectively, and assume that fi (t, ·, ·) or
−gi (t, ·, ·) is strictly convex throughout [a, b] for at least one index i ∈ k with the
corresponding component ũi of ũ (and Φ(x̃, ũ)) positive, or hj (t, ·, ·) is strictly
convex throughout [a, b] for at least one j ∈ m with the corresponding component ṽj (t) of ṽ(t) positive on [a, b]. Then x̃(t) = x ∗ (t) for all t ∈ [a, b], that
is, x̃ is an optimal solution of (P) and ϕ(x ∗ ) = ξ(z̃).
Proof. The proof is similar to that of Theorem 5.3.
We next identify two reduced versions of (DIII) and (D̃III), which are the counterparts of (DIIA) and (D̃IIA) introduced in the preceding section. These problems are obtained by dropping the constraints (6.6) and (6.7) and modifying
the remaining constraints and objective functions of (DIII) and (D̃III). They take
the following forms:
(DIIIA)
1
g
t,
y,
ẏ
− βi (t)T Bi (t)y dt
u
i
i=1 i
a
b
k
ui fi t, y, ẏ + αi (t)T Ai (t)y
×
Maximize b k
a
i=1
m
+
(6.23)
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y dt
j=1
subject to
y(a) = x a ,
y(b) = x b ,
(6.24)
4178
G. J. ZALMAI
Γ (y, u, β)
k
ui ∇2 fi t, y, ẏ + Ai (t)T αi (t)
i=1
m
T j
vj (t) ∇2 hj t, y, ẏ + Cj (t) γ (t)
+
j=1
k
− Π(y, v, α) + ∆(y, v, γ)
ui ∇2 gi t, y, ẏ − Bi (t)T βi (t)
− D Γ (y, u, β)
i=1
k
m
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
j=1
− Π(y, v, α) + ∆(y, v, γ)
k
ui ∇3 gi t, y, ẏ
= 0,
t ∈ [a, b],
i=1
(6.25)
m
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y ≥ 0,
t ∈ [a, b],
(6.26)
j=1
i ∗
i ∗
β (t)
α (t)
t ∈ [a, b], i ∈ k,
L(i) ≤ 1,
M(i) ≤ 1,
j
∗
γ (t)
t ∈ [a, b], j ∈ m,
N(j) ≤ 1,
y ∈ PWSn [a, b],
u ∈ U,
βi ∈ PWSqi [a, b],
v ∈ PWSm
+ [a, b],
i ∈ k,
(6.27)
αi ∈ PWSpi [a, b],
γ j ∈ PWSrj [a, b],
j ∈ m,
(6.28)
where Γ and Π are as defined in the description of (DIIA) and
∆(y, v, γ) =
b m
a
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y dt;
(6.29)
j=1
(D̃IIIA)
1
Maximize b k
i
T
i=1 ui gi t, y, ẏ − β (t) Bi (t)y dt
a
×
b
k
a
ui fi t, y, ẏ + αi (t)T Ai (t)y
i=1
+
m
vj (t) hj t, y, ẏ + γ j (t)T Cj (t)y dt
j=1
subject to (6.24), (6.26), (6.27), (6.28), and
Γ (y, u, β)
k
T
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
i=1
+
m
j=1
T
vj (t) ∇2 hj t, y, ẏ + γ j (t)T Cj (t)
(6.30)
4179
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
k
T
− Π(y, v, α) + ∆(y, v, γ)
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t)
i=1
− D Γ (y, u, β)
k
m
T T
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
j=1
− Π(y, v, α) + ∆(y, v, γ)
k
T
ui ∇3 gi t, y, ẏ
(x − y)
i=1
≥0
∀t ∈ [a, b], x ∈ F,
(6.31)
or
k
b T
Γ (y, u, β)
ui ∇2 fi t, y, ẏ + αi (t)T Ai (t)
a
i=1
+
m
T
j
T
vj (t) ∇2 hj t, y, ẏ + γ (t) Cj (t)
j=1
k
T
− Π(y, v, α) + ∆(y, v, γ)
ui ∇2 gi t, y, ẏ − βi (t)T Bi (t) (x − y)
+ Ψ (y, u)
i=1
k
m
T T
ui ∇3 fi t, y, ẏ +
vj (t)∇3 hj t, y, ẏ
i=1
− Π(y, v, α) + ∆(y, v, γ)
j=1
k
T ui ∇3 gi t, y, ẏ
ẋ − ẏ dt
i=1
≥ 0 ∀x ∈ F.
(6.32)
Following the patterns of Theorems 5.4, 5.5, 5.6 and Theorems 6.1, 6.2, 6.3,
one can readily state and prove similar duality results for (P)-(DIIIA) and (P)(D̃IIIA).
7. Problems containing square roots of positive semidefinite quadratic
forms. In this section, we briefly discuss an interesting special case of (P)
obtained by choosing all the norms to be the 2 -norm.
Let · L(i) , · M(i) , i ∈ k, and · N(j) , j ∈ m, be the 2 -norm · 2 and
define Pi (t) = Ai (t)T Ai (t), Qi (t) = Bi (t)T Bi (t), i ∈ k, and Rj (t) = Cj (t)T Cj (t),
j ∈ m. Then it is clear that Pi (t), Qi (t), i ∈ k, and Rj (t), j ∈ m, are n ×
n symmetric positive semidefinite matrices and, consequently, the functions
4180
G. J. ZALMAI
x(t) → [x(t)T Pi (t)x(t)]1/2 , x(t) → [x(t)T Qi (t)x(t)]1/2 , i ∈ k, and x(t) →
[x(t)T Rj (t)x(t)]1/2 , j ∈ m, are convex on Rn . With these choices of the norms
and matrices, (P), (P1), (P2), and (P3) take the following forms:
(P∗ )
b 1/2 T
dt
a fi t, x, ẋ + x Pi (t)x
Minimize max b 1/2 T
1≤i≤k
dt
a gi t, x, ẋ − x Qi (t)x
(7.1)
subject to
x(b) = x b ,
x(a) = x a ,
1/2
≤ 0, t ∈ [a, b], j ∈ m,
hj t, x, ẋ + x T Rj (t)x
n
x ∈ PWS [a, b];
(7.2)
(7.3)
(7.4)
(P∗ 1)
b 1/2 T
dt
a f1 t, x, ẋ + x P1 (t)x
Minimize b 1/2 ;
T
x∈F∗
dt
a g1 t, x, ẋ − x Q1 (t)x
(7.5)
(P∗ 2)
Minimize max
x∈F∗
b
1/2 fi t, x, ẋ + x T Pi (t)x
dt;
1≤i≤k a
(7.6)
(P∗ 3)
Minimize
x∈F∗
b
a
1/2 f1 t, x, ẋ + x T P1 (t)x
dt,
(7.7)
where F∗ is the feasible set of (P∗ ), that is,
F∗ = x ∈ PWSn [a, b] : (7.2) and (7.3) hold .
(7.8)
To see more explicitly the changes that will be required in specializing the
optimality conditions of Section 3 for (P∗ ), we next combine, modify, and restate Theorems 3.2 and 3.4 for (P∗ ).
DUALITY MODELS FOR SOME NONCLASSICAL PROBLEMS . . .
4181
Theorem 7.1. A feasible solution x ∗ of (P∗ ) is optimal if and only if there
∗i
∗i
∗j
∈ PWSn [a, b], i ∈ k,
exist u∗ ∈ U , λ∗ ∈ R+ , v ∗ ∈ PWSm
+ [a, b], ρ , σ , θ
j ∈ m, such that the following relations hold for all t ∈ [a, b]:
k
∗
∗
+ Pi (t)ρ ∗i (t) − λ∗ ∇2 gi t, x ∗ , ẋ ∗ − Qi (t)σ ∗i (t)
u∗
i ∇2 fi t, x , ẋ
i=1
+
m
vj∗ (t) ∇2 hj t, x ∗ , ẋ ∗ + Rj (t)θ ∗j (t)
j=1
−D
k
∗
∗
u∗
− λ∗ ∇3 gi t, x ∗ , ẋ ∗
i ∇3 fi t, x , ẋ
i=1
+
m
vj∗ (t)∇3 hj
∗
t, x , ẋ
∗
= 0,
j=1
m
1/2 vj∗ (t) hj t, x ∗ , ẋ ∗ + x ∗T Rj (t)x ∗
= 0,
j=1
b k
a
i=1
∗T
1/2
∗
∗
u∗
+ x Pi (t)x ∗
i fi t, x , ẋ
1/2 − λ∗ gi t, x ∗ , ẋ ∗ − x ∗T Qi (t)x ∗
dt = 0,
ρ ∗i (t)T Pi (t)ρ ∗i (t) ≤ 1,
∗j
T
σ ∗i (t)T Qi (t)σ ∗i (t) ≤ 1,
i ∈ k,
∗j
(t) ≤ 1, j ∈ m,
1/2
,
ρ ∗i (t)T Pi (t)x ∗ = x ∗T Pi (t)x ∗
∗T
∗i
T
∗
∗ 1/2
σ (t) Qi (t)x = x Qi (t)x
, i ∈ k,
1/2
θ ∗j (t)T Rj (t)x ∗ = x ∗T Rj (t)x ∗
, j ∈ m.
θ
(t) Rj (t)θ
(7.9)
In a similar manner, one can readily specialize and restate Theorems 3.5,
3.6, 3.7, (DI), (D̃I), Theorems 4.1, 4.2, 4.3, (DII), (D̃II), (DIIA), (D̃IIA), Theorems
5.1, 5.2, 5.3, 5.4, 5.5, 5.6, (DIII), (D̃III), (DIIIA), (D̃IIIA), and Theorems 6.1, 6.2,
6.3 for (P∗ ), (P∗ 1), (P∗ 2), and (P∗ 3).
Mathematical programming problems containing square roots of quadratic
forms have been the subject of numerous investigations. These problems have
arisen in stochastic programming, multifacility location problems, and portfolio selection problems among others. Many optimality and duality results for
several classes of these problems have appeared in the related literature. A
fairly long list of references pertaining to various aspects of these problems is
given in [17].
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4182
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[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
G. J. ZALMAI
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G. J. Zalmai: Department of Mathematics and Computer Science, Northern Michigan
University, Marquette, MI 49855, USA
E-mail address: gzalmai@nmu.edu