Semidefinite Programming Relaxation (SDP)

• Semidefinite programming relaxation is a convex optimization technique that transforms non-convex quadratic and polynomial problems into tractable forms using PSD constraints.
• It achieves this by lifting problems into a higher-dimensional matrix space and replacing difficult rank constraints with positive semidefiniteness.
• SDP relaxations provide effective bounds for NP-hard challenges in control, combinatorial optimization, and signal processing, enabling practical applications.

A semidefinite programming (SDP) relaxation is a convex programming technique that replaces a hard (often combinatorial or non-convex) optimization problem with an SDP whose feasible region is an intersection of a linear space and the cone of positive semidefinite (PSD) matrices. SDP relaxations are widely used to provide tractable approximations or upper/lower bounds for NP-hard quadratic programs, polynomial optimization, and a broad variety of problems in control, combinatorial optimization, statistics, and signal processing.

1. Fundamental Structure and Principle

Given a non-convex quadratic or polynomial optimization, SDP relaxations are constructed by "lifting" the original problem into a higher-dimensional space of matrices and replacing rank constraints with PSD constraints. The archetypal form is

$$\begin{aligned} \min_{X\in \mathbb{S}^n} \quad & \langle C, X \rangle \ \text{s.t.} \quad & \langle A_i, X \rangle = b_i,\quad i=1,\ldots,m \ & X \succeq 0, \end{aligned}$$

where $\langle A, B \rangle = \mathrm{Tr}(A^T B)$, $A_i, C \in \mathbb{S}^n$, and $X \succeq 0$ imposes positive semidefiniteness on the matrix variable $X$ (Shah et al., 2016).

The mechanism is to relax a quadratic form $x^T Q x$ for $x \in \mathbb{R}^n$ into $\mathrm{Tr}(QX)$ with $X \succeq 0$ and $X$ approximating $\langle A, B \rangle = \mathrm{Tr}(A^T B)$0. The non-convex constraint $\mathrm{rank