PFA
1. Consider the function ?: ℝ2 → ℝ defined by ?(?) = − 2x1 2 + 5x1x2 − 4x2 2. (a) Find the first derivative ??(?) and the second ?2?(?) of f, and using the characterization of concave functions twice differentiable, verify that f is concave. (Hint: to show that a symmetric matrix is semi-defined negative you can use the fact that the condition equates to the matrix eigenvalues being non-positive.) (b) Consider the function �̂�: ℝ2 → ℝ4 defined by �̂�(?) = ( −?1 −?2 ?1 − 1 ?2 − 1 ) Defining ? = {? ? ℝ2: �̂�?(?) ≤ 0, ? = 1,2,3,4}, check that ? is compact and convex, and find the solution to the problem of ?????ℝ2?(?) subject to ? ? ?. ________________________________________________________________________ 2. ?: ? → ℝ is a function of N variables that is differentiable, with domain ? ⊆ ℝ? convex, ? ≠ 0. a) ??(?) = {ℎ ? ℝ ?: exists �̅� > 0 such that ? + ?ℎ ? ?, for all 0 < ? ≤ �̅�} the set of feasible directions in c ? d. prove that ?d(c) = {?(x − c): x ? d, ? ≥ 0} b) in the particular case where d = r+ n, suppose x* solve the problem ???????(?). check, based on first order conditions in terms of viable directions, that for all i = 1,....,n, worth that: ??(?∗) ??? ≤ 0, ?? ∗ ??(?∗) ??? = 0,?? ∗ ≥ 0. (hint: try to characterize the set first ??(? ∗) defined in the previous letter in the case where ? = ?+ ?) c) ? ? ?+ ? and ? ? ℝ++. consider the function ?: ? → ℝ set on ? = ?+ ? by means of ?(?) = − 1 〈?,?〉+? . prove that f is a concave function and that there is no solution to the optimization problem ???????(?). ≤="" �̅�}="" the="" set="" of="" feasible="" directions="" in="" c="" d.="" prove="" that="" d(c)="{?(x" −="" c):="" x="" d,="" ≥="" 0}="" b)="" in="" the="" particular="" case="" where="" d="R+" n,="" suppose="" x*="" solve="" the="" problem="" (?).="" check,="" based="" on="" first="" order="" conditions="" in="" terms="" of="" viable="" directions,="" that="" for="" all="" i="1,....,N," worth="" that:="" (?∗)="" ≤="" 0,="" ∗="" (?∗)="" =="" 0,??="" ∗="" ≥="" 0.="" (hint:="" try="" to="" characterize="" the="" set="" first="" (?="" ∗)="" defined="" in="" the="" previous="" letter="" in="" the="" case="" where="" =="" +="" )="" c)="" +="" and="" ℝ++.="" consider="" the="" function="" :="" →="" ℝ="" set="" on="" =="" +="" by="" means="" of="" (?)="−" 1="" 〈?,?〉+?="" .="" prove="" that="" f="" is="" a="" concave="" function="" and="" that="" there="" is="" no="" solution="" to="" the="" optimization="" problem="">