1. CHAPTER 12 STATISTICAL METHODS FOR OPTIMIZATION IN DISCRETE PROBLEMS
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

2. 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)

3. 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

4. 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

5. 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 
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

6. Example: Widths of 95% Acceptance Intervals Increasing with K in Tukey–Kramer Test (n1=n2=…=nK=10)

7. 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  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

8. 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 = 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 < 0 for observed differences mj
To show that L(m) < L(j) for all j requires additional analysis

9. Example of Many-to-One Test with Known Variances (Example 12
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 , , and for acceptance regions
Valid to take
Under Bonferroni/Chebyshev:
Under Bonferroni/normal noise:
Under Slepian/normal noise:
Note tighter (smaller) acceptance regions when assuming normal noise

10. Widths of 95% Acceptance Intervals (< 0) for Tukey-Kramer and Many-to-One Tests (n1=n2=…=nK=10)

11. 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)