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The instructions and questions for a m.phil. Exam in statistical science, focusing on optimization and mathematics of operational research. The exam includes questions on topics such as lagrangian functions, lagrangian sufficiency theorem, the simplex algorithm, the ford-fulkerson algorithm, and the max-flow min-cut theorem. Students are required to solve optimization problems, use the simplex algorithm and ford-fulkerson algorithm, and prove mathematical theorems.
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Monday, 31 May, 2010 1:30 pm to 4:30 pm
Attempt no more than FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
1 Consider the optimization problem
Minimize f (x) subject to h(x) = b over x ∈ X ,
where X ⊂ Rn^ and b ∈ Rm. Define the Lagrangian function for this problem and then state and prove the Lagrangian Sufficiency Theorem. Define the function φ by
φ(b) = inf x∈X
{f (x) : h(x) = b}.
Define the Strong Lagrangian property and show that the following are equivalent:
(a) there exists a non-vertical supporting hyperplane to φ at b;
(b) the problem is Strong Lagrangian.
Minimize f =
vi x− i 1 in x > 0 subject to
ai xi 6 b where ai, vi > 0 for all i and b > 0. [In this example f is the variance of an estimate derived from a stratified sample survey subject to a cost constraint: xi is the size of the sample for the ith^ stratum, the ai and vi are measures of sampling cost and of variability for this stratum.]
Check that the change in the minimal variance f for a small change δb in available resources is λδb where λ is the Lagrange multiplier.
Mathematics of Operational Research
(a) State and prove the Max-Flow Min-Cut Theorem.
(b) Use the Ford-Fulkerson Algorithm to calculate a maximum flow between a source at node 1 and a destination at node 8 in the following network.
Here the number by each arc represents the capacity of that arc. What is the min-cut of this network?
(c) Starting from the given feasible solution, minimize the cost of flows in the transportation problem given by the following tableau.
[Note: In this tableau the circled numbers indicate an initial feasible set of non-zero flows, the numbers in the squares are the costs, the numbers to the right of the tableau are supplies and the numbers below are demands.]
Mathematics of Operational Research
Consider the Boolean formula with N clauses
(x 11 ∨ x 12 ∨... ∨ x 1 M 1 ) ∧ (x 21 ∨ x 22 ∨... ∨ x 2 M 2 ) ∧... ∧ (xN 1 ∨... V xN MN ) (1)
where xij ∈ {X 1 ,... , XK } ∪ { X¯ 1 ,... , X¯K }. Here ∧ means “AND” and ∨ means “OR” and X¯ means “NOT X”.
The SAT problem considers the assignment of variables, Xi ∈ {true, false} , i = 1, 2 ,... , K such that (1) is true.
The MAX-SAT problem considers the assignment of variables such that the maxi- mum number of clauses in (1) are true.
Express the SAT problem as an integer linear program.
Express the MAX-SAT problem as an integer linear program.
Consider the following approximation algorithm for MAX-SAT.
Greedy: Pick the variable z ∈ {X 1 ,... , XK } ∪ { X¯ 1 ,... , X¯K } that occurs in the largest number of clauses in (1). Set z true and ¯z false. This reduces formula (1) to an expression on K − 1 variables. Repeat until no variables remain.
Show that Greedy is a 12 -approximation of MAX-SAT.
[Recall: Algorithm H with solution αH is an ǫ-approximation to a maximization problem with optimal solution α∗^ if for all problem instances,
αH > (1 − ǫ)α∗. ]
Mathematics of Operational Research [TURN OVER
Let X be a convex set of strategies. Recall that a strategy x∗^ ∈ X is an evolutionary stable strategy (ESS) if for every y ∈ X, y 6 = x∗^ then
e(x∗, ¯x) > e(y, x¯)
where ¯x = (1 − ǫ)x∗^ + ǫy for sufficiently small ǫ > 0. Briefly discuss the interpretation of e(. , .), x∗^ and y in this definition.
Show that a strategy x∗^ is an ESS if and only if for every y ∈ X, y 6 = x∗
e(x∗, x∗) > e(y, x∗)
and if e(x∗, x∗) = e(y, x∗) then e(x∗, y) < e(y, y).
Suppose that a strategy x ∈ X is a mixture (p, 1 − p) of the two pure strategies “Hawk” = (1, 0) and “Dove” = (0, 1) and that for the pure strategies the pay off matrix is
Hawk Dove Hawk 12 (V − D) V Dove 0 12 V
Find an ESS when
(i) V > D ,
(ii) V = D ,
(iii) V < D ,
justifying your answer in each case.
Mathematics of Operational Research
