Optimization — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Ques. 1 (i Examine the converty of the function f71,22) = 4x} +*3+4x1x2. (a) (ii) Over what domain is the function f(x) = x2(x2 - 1) convex? (b) Let f1. f2,.., fk: RT → R be convex functions. Consider the function f defined by f(x) = max/i (x), f2(x),..., fk (x)). Show that f is convex. (c) Define the epigraph of a function. Show that the epigraph of the function f(x) = +(1 - x2), 0 ≤ x ≤ 1 is convex? Justify your answer. Ques.2 Let S be a nonempty convex set in R" and let f:S → R be a convex function. Consider (a) the problem to minimize f(x) subject to x € S. Suppose that * E S is a local optimal solution to the problem. Then show that x is a global optimal solution. Further prove that, if f is strictly convex, then i is the unique global optimal solution. (b) Define a quasiconvex, quasiconcave and quasimonotone functions. State z. necessary and sufficient characterization of a quasiconvex function in terms of its lower level sets. (c) Give example of the following (i) a quasiconvex function which is not pseudoconvex, (ii) a pseudoconvex function which is not convex. Ques.3 Suppose that f: R" → R is twice differentiable at x. If i is a local minimum, then show (a) that f(x) = 0 and H (x) is positive semidefinite . (b) Consider the univariate function f(x) = xe-x. Find all local minima/maxima and inflection points. Also, what can you claim about a global minimum and a global maximum forf? ( c) Consider the following problem: Verify that the Fritz John conditions hold true at the optimal point (1,0). Also find the Lagrange multipliers such that (1 ,0) becomes the Fritz John point. Ques.4 Consider the following problem Find a point satisfying the KKT conditions and verify that it is an optimal solution. (b) Let x be a feasible solution to the problem (P) defined as: Subject to g/(x) ≤ 0, for i = 1,2,...m hi(x) = 0, for i = 1,2,.../ Where X is a nonempty open set in R"; f:R* → R; giR" → Rfori= Let (u, v) be a feasible solution to its dual (D) with objective value 0(1, v). Prove that f(x) ≥ 0(u, v). ( c) Consider the following primal problem Where X = {(0,0), (0,4), (4,4), (4,0), (1,2), (2,1)). Find the clual objective function. wither find the 0) c Further find the optimal solutions of primal and dual problems and show that there exists a duality gap. Ques.5 Consider the following Quadratic Programming Problem (QPP): Express the objective in the standard QPP form c"x + x" Dx. Is D positive definite? Write the KKT conditions for this problem. (b) Use Wolfe's Method to solve the following Quadratic Programming Problem: (c) Starting from the point (0,0), perform two iterations of the steepest descent method to minimizef(x1,22) = 2x] + 2x1x2+ x3 + x1- x2 over (x1"x=) E R2. Is the obtained solution after two iterations optimal? Ques.6 Use Newton's method to minimize*122) =2x} + 2xyx2 + 3x3, (x1,x2) E R2 (a) starting with the point (-2, -3) (b) Consider the linear fractional programming problem Maximize aT×+B subject to Ax = b x ≥ 0.