Convex Optimization · Problem 4.41

Duality — Boyd & Vandenberghe

Mathematics — Optimization · lmi-tests-copositive-p0-matrices

Problem

LMI tests for copositive matrices and

P_0

-matrices. A matrix

A \in \mathbf{S}^n

is said to be copositive if

x^T A x \geq 0

for all

x \succeq 0

(see exercise 2.35). A matrix

A \in \mathbf{R}^{n \times n}

is said to be a

P_0

-matrix if

\max_{i=1,\ldots,n} x_i(Ax)_i \geq 0

for all

x

. Checking whether a matrix is copositive or a

P_0

-matrix is very difficult in general. However, there exist useful sufficient conditions that can be verified using semidefinite programming. (a) Show that

A

is copositive if it can be decomposed as a sum of a positive semidefinite and an elementwise nonnegative matrix:

A = B + C, \quad B \succeq 0, \quad C_{ij} \geq 0, \quad i, j = 1, \ldots, n. \tag{4.71}

Express the problem of finding

B

and

C

that satisfy (4.71) as an SDP feasibility problem. (b) Show that

A

is a

P_0

-matrix if there exists a positive diagonal matrix

D

such that

DA + A^T D \succeq 0. \tag{4.72}

Express the problem of finding a

D

that satisfies (4.72) as an SDP feasibility problem.

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