Slides for Introduction to Stochastic Search
and Optimization (ISSO) by J. C. Spall
CHAPTER 12
STATISTICAL METHODS FOR
OPTIMIZATION IN DISCRETE PROBLEMS
•Organization of chapter in ISSO
–Basic problem in multiple comparisons
•Finite number of elements in search domain 
–Tukey-Kramer test
–“Many-to-one” tests for sharper analysis
•Measurement noise variance known
•Measurement noise variance unknown (estimated)
–Ranking and selection methods
Background
• Statistical methods used here to solve optimization
problem
– Not just for evaluation purposes
• Extending standard pairwise t-test to multiple
comparisons
• Let     {1, 2, …, K} be finite search space (K
possible options)
• Optimization problem is to find the j such that  = j
• Only have noisy measurements of L(i)
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Applications with Monte Carlo
Simulations
• Suppose wish to evaluate K possible options in a
real system
– Too difficult to use real system to evaluate options
• Suppose run Monte Carlo simulation(s) for each of
the K options
• Compare options based on a performance measure
(or loss function) L() representing average (mean)
performance
–  represents options that can be varied
– Monte Carlo simulations produce noisy
measurement of loss function L at each option
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Statistical Hypothesis Testing
• Null hypothesis: All options in   {1, 2, …, K}
are effectively the same in the sense that L(1) =
L(2) = … = L(K)
• Challenge in multiple comparisons: alternative
hypothesis is not unique
– Contrasts with standard pairwise t-test
• Analogous to standard t-test, hypothesis testing
based on collecting sample values of L(1), L(2),
and L(K), forming sample means L1, L2 ,..., LK
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Tukey–Kramer Test
• Tukey (1953) and Kramer (1956) independently
developed popular multiple comparisons analogue to
standard t-test
• Recall null hypothesis that all options in   {1, 2, …,
K} are effectively the same in the sense that L(1) =
L(2) = … = L(K)
• Tukey–Kramer test forms multiple acceptance
intervals for K(K–1)/2 differences ij  Li  L j
– Intervals require sample variance calculation based on
samples at all K options
• Null hypothesis is accepted if evidence suggests all
differences ij lie in their respective intervals
– Null hypothesis is rejected if evidence suggests at least
one ij lies outside its respective interval
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Example: Widths of 95% Acceptance
Intervals Increasing with K in Tukey–Kramer
Test (n1=n2=…=nK=10)
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Example of Tukey–Kramer Test
(Example 12.2 in ISSO)
• Goal: With K = 4, test null hypothesis L(1) = L(2) =
L(3) = L(4) based on 10 measurements at each i
• All (six) differences ij  Li  L j must lie in acceptance
intervals [–1.23, 1.23]
• Find that 34 = 1.72
– Have 34  [–1.23, 1.23]
• Since at least one ij is not in acceptance interval, reject
null hypothesis
• Conclude at least one i likely better than others
– Further analysis required to find i that is better
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Multiple Comparisons Against One Candidate
• Assume prior information suggests one of K points is
optimal, say m
• Reduces number of comparisons from K(K–1)/2 differences
ij = Li  L j to only K–1 differences mj
• Under null hypothesis, L(m)  L(j) for all j
• Aim to reject null hypothesis
– Implies that L(m) < L(j) for at least some j
• Tests based on critical values mj < 0 for observed
differences mj
• To show that L(m) < L(j) for all j requires additional
analysis
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Example of Many-to-One Test with
Known Variances (Example 12.3 in ISSO)
• Suppose K = 4, m = 2  Need to compute 3 critical
values 21 , 23, and 24 for acceptance regions { 2 j  2 j }
• Valid to take   21  23  24
• Under Bonferroni/Chebyshev:    3.96
• Under Bonferroni/normal noise:    1.10
• Under Slepian/normal noise:    1.09
• Note tighter (smaller) acceptance regions when assuming
normal noise
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Widths of 95% Acceptance Intervals (< 0)
for Tukey-Kramer and Many-to-One Tests
(n1=n2=…=nK=10)
Ranking and Selection:
Indifference Zone Methods
• Consider usual problem of determining best of K possible
options, represented 1 , 2 ,…, K
• Have noisy loss measurements yk(i )
• Suppose analyst is willing to accept any i such that L(i)
is in indifference zone [L(), L() + )
• Analyst can specify  such that
P(correct selection of  = )  1 
whenever L(i) L()   for all i  
• Can use independent sampling or common random
numbers (see Section 14.5 of ISSO)
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