# MAT3007 · Homework 6 solution

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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