Question: Consider the optimization problems (no assumption of convexity here) 1. Prove that p* 1 p* 2 (i.e., enlarging the feasible set cannot worsen the

Consider the optimization problems (no assumption of convexity here)

P = pi P2 P3 = 13 P23 = min fo(x) XEX

min fo(x) XEX min f(x) xeting min fo(x), xen3

1. Prove that p*1 ≥  p*2 (i.e., enlarging the feasible set cannot worsen the optimal objective).
2. Prove that, if p*1 = p*2, then it holds that

3. Assume that all problems above attain unique optimal solutions. Prove that, under such hypothesis, if p*1 = p*2, then it holds that

P = pi P2 P3 = 13 P23 = min fo(x) XEX min fo(x) XEX min f(x) xeting min fo(x), xen3

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