By Gordon Whyburn

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In this case the set {y E R " IIIX yll ::; I} is a minimum volume ellipsoid (center ed at the origin) containing the vectors yl , y2, .. ) (d) Show the Man gasari an -Fromovitz cons t raint qu alification holds at X by considering t he direction d = - X. (e) Wri te down t he Karu sh-Kuhn-Tucker condit ions that X must sat isfy. (f) When {yl , y2, . . , yrn } is the st andard basis of R n, t he opt imal solution of t he probl em in part (c) is X = I . Find t he correspo nd ing Lagran ge mult iplier vector.

Line a r inde p endence implie s Mangasarian-Fromov itz) If the set of vectors {ai , a 2 , ... , am } in E is linearly independent , prove directly there exists a direction d in E satisfying (ai , d) < 0 for i = 1,2, .. , m. 4. For each of the following problems , explain why there must exist an optimal solution, and find it by using the Karush-Kuhn-Tucker conditions. (a) inf subj ect to xI + X~ X2 + 10 < 0 -2Xl - < O. - Xl (b) inf subject to + 6x~ 5xI Xl 25 - 4 < 0 xI - x~ < o. 5. (Cauchy-Schwarz a n d steepest descent ) For a nonzero vector y in E , use the Karush-Kuhn-Tucker conditions to so lve the problem inf{ (y, X) I IIx I1 2 ::; I}.

B) C = B , the unit ball. (c) C a subspace. (d) C a closed halfspace: {x bE R. (e) C = {x ERn I x j ;:: I (a , x l :::; b} where 0 =I- a E E and 0 for all j E J} (for J C {I , 2, . . , n} ). 3. (Self-dual cones) Prove each of the following cones K satisfy the relationship NK(O) = -K . 1 Optimality Conditions 19 4. (Normals to a ffine sets) Given a linear map A : E ----? Y (where Y is a Euclidean space) and a point b in Y , prove the normal cone to t he set {x E E l A x = b} at any point in it is A*Y .

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Topological Analysis, First Edition by Gordon Whyburn

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