7.23
MATLAB Solution of Constrained Optimization Problems
Step 3: Invoke constrained optimization program (write this in new MATLAB file).
clc
clear all
warning off
x0 = [.1,.1, 3.0]; % Starting guess
fprintf ('The values of function value and constraints at
starting pointn');
f=objfun (x0)
[c, ceq] = constraints (x0)
options = optimset ('LargeScale', 'off');
[x, fval]=fmincon (@objfun, x0, [], [], [], [], [], [],
@constraints, options)
fprintf ('The values of constraints at optimum solutionn');
[c, ceq] = constraints (x)
% Check the constraint values at x
This Produces the Solution or Ouput as follows:
The values of function value and constraints at
starting point
f=
4.0410
c=
-8.9800
-5.0200
-2.0000
-0.1000
-0.1000
-3.0000
ceq =
[]
Optimization terminated: first-order optimality measure
less
than options. TolFun and maximum constraint violation is
less
than options.TolCon.
Active inequalities (to within options.TolCon = 1e-006):
lower upper ineqlin
1
2
4
x=
0 1.4142 1.4142
fval =
1.4142
ineqnonlin
The values of constraints at optimum solution
c=
475
476
Nonlinear Programming III: Constrained Optimization Techniques
-0.0000
-0.0000
-3.5858
0
-1.4142
-1.4142
ceq =
[]
REFERENCES AND BIBLIOGRAPHY
7.1
R. L. Fox, Optimization Methods for Engineering Design, Addison-Wesley, Reading, MA,
1971.
7.2
M. J. Box, A new method of constrained optimization and a comparison with other
methods, Computer Journal, Vol. 8, No. 1, pp. 42-52, 1965.
7.3
E. W. Cheney and A. A. Goldstein, Newton's method of convex programming and
Tchebycheff approximation, Numerische Mathematik, Vol. 1, pp. 253-268, 1959.
7.4
J. E. Kelly, The cutting plane method for solving convex programs, Journal of SIAM ,
Vol. VIII, No. 4, pp. 703-712, 1960.
7.5
G. Zoutendijk, Methods of Feasible Directions, Elsevier, Amsterdam, 1960.
7.6
W.W. Garvin, Introduction to Linear Programming, McGraw-Hill, New York, 1960.
7.7
S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems, Prentice Hall, Englewood Cliffs, NJ, 1972.
7.8
G. Zoutendijk, Nonlinear programming: a numerical survey, SIAM Journal of Control
Theory and Applications, Vol. 4, No. 1, pp. 194-210, 1966.
7.9
J. B. Rosen, The gradient projection method of nonlinear programming, Part I: linear
constraints, SIAM Journal, Vol. 8, pp. 181-217, 1960.
7.10
J. B. Rosen, The gradient projection method for nonlinear programming, Part II: nonlinear
constraints, SIAM Journal, Vol. 9, pp. 414-432, 1961.
7.11
G. A. Gabriele and K. M. Ragsdell, The generalized reduced gradient method: a reliable
tool for optimal design, ASME Journal of Engineering for Industry, Vol. 99, pp. 384-400,
1977.
7.12
M. J. D. Powell, A fast algorithm for nonlinearity constrained optimization calculations,
in Lecture Notes in Mathematics, G. A. Watson et al., Eds., Springer-Verlag, Berlin,
1978.
7.13
7.14
M. J. Box, A comparison of several current optimization methods and the use of
transformations in constrained problems, Computer Journal, Vol. 9, pp. 67-77, 1966.
7.15
C. W. Carroll, The created response surface technique for optimizing nonlinear restrained
systems, Operations Research, Vol. 9, pp. 169-184, 1961.
7.16
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained
Minimization Techniques, Wiley, New York, 1968.
7.17
W. I. Zangwill, Nonlinear programming via penalty functions, Management Science, Vol.
13, No. 5, pp. 344-358, 1967.
A. V. Fiacco and G. P. McCormick, Extensions of SUMT for nonlinear programming: equality constraints and extrapolation, Management Science, Vol. 12, No. 11,
pp. 816-828, July 1966.
References and Bibliography
477
7.18
D. Kavlie and J. Moe, Automated design of frame structure, ASCE Journal of the Structural Division, Vol. 97, No. ST1, pp. 33-62, Jan. 1971.
7.19
J. H. Cassis and L. A. Schmit, On implementation of the extended interior penalty function, International Journal for Numerical Methods in Engineering, Vol. 10, pp. 3-23,
1976.
7.20
R. T. Haftka and J. H. Starnes, Jr., Application of a quadratic extended interior
7.21
penalty
function for structural optimization, AIAA Journal, Vol. 14, pp. 718-728, 1976.
7.22
A. V. Fiacco and G. P. McCormick, SUMT Without Parmaeters, System Research Memorandum 121, Technical Institute, Northwestern University, Evanston, IL, 1965.
7.23
A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.
7.24
R. T. Rockafellar, The multiplier method of Hestenes and Powell applied to convex programming, Journal of Optimization Theory and Applications, Vol. 12, No. 6, pp. 555-562,
1973.
7.25
B. Prasad, A class of generalized variable penalty methods for nonlinear programming,
Journal of Optimization Theory and Applications, Vol. 35, pp. 159-182, 1981.
7.26
L. A. Schmit and R. H. Mallett, Structural synthesis and design parameter hierarchy,
Journal of the Structural Division, Proceedings of ASCE , Vol. 89, No. ST4, pp. 269-299,
1963.
7.27
J. Kowalik and M. R. Osborne, Methods for Unconstrained Optimization Problems, American Elsevier, New York, 1968.
7.28
N. Baba, Convergence of a random optimization method for constrained optimization
problems, Journal of Optimization Theory and Applications, Vol. 33, pp. 451-461, 1981.
7.29
J. T. Betts, A gradient projection-multiplier method for nonlinear programming, Journal
of Optimization Theory and Applications, Vol. 24, pp. 523-548, 1978.
7.30
J. T. Betts, An improved penalty function method for solving constrained parameter optimization problems, Journal of Optimization Theory and Applications, Vol. 16, pp. 1-24,
1975.
7.31
W. Hock and K. Schittkowski, Test examples for nonlinear programming codes, Journal
of Optimization Theory and Applications, Vol. 30, pp. 127-129, 1980.
7.32
J. C. Geromel and L. F. B. Baptistella, Feasible direction method for large-scale nonconvex programs: decomposition approach, Journal of Optimization Theory and Applications,
Vol. 35, pp. 231-249, 1981.
7.33
D. M. Topkis, A cutting-plane algorithm with linear and geometric rates of convergence,
Journal of Optimization Theory and Applications, Vol. 36, pp. 1-22, 1982.
7.34
M. Avriel, Nonlinear Programming: Analysis and Methods, Prentice Hall, Englewood
Cliffs, NJ, 1976.
7.35
H. W. Kuhn, Nonlinear programming: a historical view, in Nonlinear Programming,
SIAM-AMS Proceedings, Vol. 9, American Mathematical Society, Providence, RI, 1976.
7.36
J. Elzinga and T. G. Moore, A central cutting plane algorithm for the convex programming
problem, Mathematical Programming, Vol. 8, pp. 134-145, 1975.
7.37
V. B. Venkayya, V. A. Tischler, and S. M. Pitrof, Benchmarking in structural optimization,
Proceedings of the 4th AIAA/USAF/NASA/OAI Symposium on Multidisciplinary Analysis
and Optimization, Sept. 21-23, 1992, Cleveland, Ohio, AIAA Paper AIAA-92-4794.
7.38
W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture
Notes in Economics and Mathematical Systems, No. 187, Springer-Verlag, Berlin, 1981.
S. S. Rao, Multiobjective optimization of fuzzy structural systems, International Journal
for Numerical Methods in Engineering, Vol. 24, pp. 1157-1171, 1987.
478
Nonlinear Programming III: Constrained Optimization Techniques
7.39
K. M. Ragsdell and D. T. Phillips, Optimal design of a class of welded structures
using geometric programming, ASME Journal of Engineering for Industry, Vol. 98,
pp. 1021-1025, 1976.
7.40
J. Golinski, An adaptive optimization system applied to machine synthesis, Mechanism
and Machine Synthesis, Vol. 8, pp. 419-436, 1973.
7.41
H. L. Li and P. Papalambros, A production system for use of global optimization knowledge, ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 107,
pp. 277-284, 1985.
7.42
M. Avriel and A. C. Williams, An extension of geometric programming with application
in engineering optimization, Journal of Engineering Mathematics, Vol. 5, pp. 187-194,
1971.
7.43
G. A. Gabriele and K. M. Ragsdell, Large scale nonlinear programming using the generalized reduced gradient method, ASME Journal of Mechanical Design, Vol. 102, No. 3,
pp. 566-573, 1980.
7.44
A. D. Belegundu and J. S. Arora, A recursive quadratic programming algorithm with
active set strategy for optimal design, International Journal for Numerical Methods in
Engineering, Vol. 20, No. 5, pp. 803-816, 1984.
7.45
G. A. Gabriele and T. J. Beltracchi, An investigation of Pschenichnyi's recursive quadratic
programming method for engineering optimization, ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, pp. 248-253, 1987.
7.46
F. Moses, Optimum structural design using linear programming, ASCE Journal of the
Structural Division, Vol. 90, No. ST6, pp. 89-104, 1964.
7.47
S. L. Lipson and L. B. Gwin, The complex method applied to optimal truss configuration,
Computers and Structures, Vol. 7, pp. 461-468, 1977.
7.48
G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design with
Applications, McGraw-Hill, New York, 1984.
7.49
T. F. Edgar and D. M. Himmelblau, Optimization of Chemical Processes, McGraw-Hill,
New York, 1988.
7.50
A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, Engineering Optimization Methods
and Applications, 2nd ed., Wiley, New York, 2006.
7.51
L. S. Lasdon, Optimization Theory for Large Systems, Macmillan, New York, 1970.
7.52
R. T. Haftka and Z. G¨urdal, Elements of Structural Optimization, 3rd ed., Kluwer Academic, Dordrecht, The Netherlands, 1992.
REVIEW QUESTIONS
7.1
Answer true or false:
(a) The complex method is similar to the simplex method.
(b) The optimum solution of a constrained problem can be the same as the unconstrained
optimum.
(c) The constraints can introduce local minima in the feasible space.
(d) The complex method can handle both equality and inequality constraints.
(e) The complex method can be used to solve both convex and nonconvex problems.
(f) The number of inequality constraints cannot exceed the number of design variables.
(g) The complex method requires a feasible starting point.