## Description

Problem 1 (25pts). Consider the following function:

f(x, y, z) = 3x

2 − 2x − 2xy + 3y

2 − 2y − 2zy + 3z

2 − 2z − 2xz

(a). Considering the 1st-order necessary condition, try to find the candidate minimizers of f(x, y, z).

(b). Considering the 2nd-order sufficient condition, whether these candidates are indeed local minimizers?

(c). Is (0,0,0) a local minimizer? Why?

Problem 2 (25pts). Given a symmetric matrix A ∈ Rn×n

, consider the following problem:

minx∈Rn x

T Ax

subject to 2 − x

T x = 0

(a). Give the KKT conditions of this problem.

(b). If A is positive definite without repeated eigenvalues, how many different KKT points are there

at most?

(c). If A is positive definite without repeated eigenvalues, what is the minimum value of this problem, and how many local minimizers? (Hint:According to Rayleigh quotient, min{x

T Ax/(x

T x)} =

λmin, where λmin is the minimum eigenvalue of A)

Problem 3 (25pts). Construct the KKT conditions for the following linear program:

maximize 3×1 + x2 + 4×3

subject to x1 + 3×2 + x3 ≤ 5

x1 + 2×2 + 2×3 ≤ 8

x1, x2, x3 ≥ 0

Problem 4 (25pts). Construct the KKT conditions for the following nonlinear program:

minimize x1ln(x1) + (x2 − 2)2 + x3

subject to x1 + x2 ≤ 3

x3 − x

2

2 ≥ 3

x1, x2, x3 ≥ 0