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Questions tagged [semidefinite-programming]

This tag is for questions regarding semidefinite programming (SDP) which is a subfield of convex optimization concerned with the optimization of a linear objective function (an objective function is a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

(Wikipedia) Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems.

In automatic control theory, SDPs are used in the context of linear matrix inequalities. SDPs are in fact a special case of cone programming and can be efficiently solved by interior point methods.

All linear programs can be expressed as SDPs, and via hierarchies of SDPs the solutions of polynomial optimization problems can be approximated.

Semidefinite programming has been used in the optimization of complex systems. In recent years, some quantum query complexity problems have been formulated in term of semidefinite programs.

402 questions
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Convert Semidefinite program forms

How do I convert the following SDP problem (written in the standard inequality form): $$\min c^T x$$ $$\text{s.t. }F(x)\succeq0$$ When $F(x)\equiv F_{0}+\sum_{i=1}^{m}x_{i}F_{i}$ when $F_{i}\in S^{n}$, $i=0,\ldots,m$ To the following conic…
Noa Haas
  • 171
13
votes
2 answers

A matrix inequality involving trace norm of a matrix and its inverse

Let $A,B \succeq 0$ be two positive semidefinite matrices. Can we get a closed form expression for the following quantity? $$ \inf_{X \succ 0} \mathrm{tr}(XA) + \mathrm{tr}(X^{-1}B) $$ We assume all matrices involved are symmetric.
O. Richard
  • 365
11
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0 answers

Analytical solution for a neat semidefinite program (SDP)

Let $A \in S^{n}_{+}$ be a positive semi-definite matrix with all entries being non-negative. I wonder if there is an analytical solution to the following SDP in correlation matrix $X \in S^{n}_{+}$ $$\begin{array}{ll} \underset{X \in…
zxzx179
  • 1,577
10
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1 answer

Writing a convex quadratic program (QP) as a semidefinite program (SDP)

Given a convex quadratic program (QP) $$\begin{array}{ll} \underset{x}{\text{minimize}} & \mathrm x^\top \mathrm Q \, \mathrm x + \mathrm r^{\top} \mathrm x + s\\ \text{subject to} & \mathrm A \mathrm x \leq \mathrm b\end{array}$$ how can one write…
Rodrigo de Azevedo
  • 23,223
9
votes
2 answers

Adding Elements to Diagonal of Symmetric Matrix to Ensure Positive Definiteness.

I have a symmetric matrix $A$, which has zeroes all along the diagonal i.e. $A_{ii}=0$. I cannot change the off diagonal elements of this matrix, I can only change the diagonal elements. I need this matrix to be positive definite. One way to ensure…
Mael
  • 793
7
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3 answers

Can every semidefinite program be solved in polynomial time?

I am reading a book about semidefinite programming that states the following: Every semidefinite program can be solved in polynomial time, up to desired accuracy $\epsilon$. Is this true? And how we can do it?
user522529
7
votes
1 answer

Spectral norm minimization via semidefinite programming

Given symmetric matrices $A_0, A_1, \dots, A_n \in \mathbb R^{m \times m}$, let $A(x) := A_0 + x_1 A_1 +\cdots + x_n A_n$. How to formulate the following unconstrained spectral minimization problem as a semidefinite program? $$\min_{x \in \mathbb…
Marcus118
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6
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1 answer

Recover primal solution from dual for matrix completion

Consider the following primal/dual SDPs $$ \min\limits_X \; \lVert X \rVert_* : \mathcal{A}(X) = b \qquad \max\limits_z \; b^T z : \lVert \mathcal{A}^*(z) \rVert \leq 1 $$ where $\lVert X \rVert_* = \sum_{i=1}^{\text{rank}(X)} \sigma_i(X)$ is the…
steve
  • 767
6
votes
2 answers

Quadratic equality constraints via SDP

I want to know if it is possible to solve a QCQP problem with quadratic equality constraints in SDP. I know it is possible to convert a QCQP to an SDP by using the Shur complement. The following worked for me thus…
Kirillvh
  • 193
6
votes
2 answers

Dual of a semidefinite program

How do I write the dual of the following semidefinite program? \begin{align} \max_{\lambda,y_i}~&\lambda \\ &\sum_{i=1}^{L}y_i\mathbf{C}_i-\lambda\mathbf{I}\geq 0 \\ &\sum_{i=1}^{L}y_i=1 \\ &y_i\geq 0 \end{align} EDIT: This is not a homework. This…
dineshdileep
  • 9,113
6
votes
1 answer

Dual of a semidefinite program in non-standard form

I have a problem with calculating the dual problem of : $$ \mbox{Minimize } tr(Y) + \frac{1}{\eta} tr(Z) $$ $$ \begin{pmatrix} Y & X \\ X & Z+\varepsilon I \end{pmatrix} \succeq 0 \mbox{, } % \begin{pmatrix} I & X \\ X & Z \end{pmatrix} …
Aymane Fihadi
  • 757
6
votes
1 answer

SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' results. Longer version I've been trying to reproduce…
user73053
  • 61
6
votes
1 answer

Characterizing duals of cones that are linear images of the positive semidefinite cone

Let $M_n$ denote the space of $n\times n$ matrices over complex numbers. The space of self-adjoint matrices is denoted $$ M_n^{sa} = \{A\in M_n\, :\, A^*=A \}, $$ where $A^*$ denotes the conjugate transpose of $A$, and the cone of positive…
Luftbahnfahrer
  • 1,992
6
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1 answer

Existence of a positive semidefinite matrix that satisfies a set of equality constraints

Given vectors $a_1, b_2, a_2, b_2 \in \mathcal{R}^{n\times 1}$, I am interested in finding a positive semi-definite matrix $M \in \mathcal{R}^{n\times n}$, $M \succeq 0$, such that $M\cdot a_1 = b_1$, $M\cdot a_2 = b_2$. Here $n \gg 2$, say $n =…
Alex Allen
  • 61
5
votes
1 answer

Relation between rank of a symmetric positive semi-definite matrix and its number of non-zero eigen values (or singular values)

Is there any relation between the rank of a symmetric positive semi-definite matrix and its number of non-zero eigenvalues (or singular values)? For a matrix $\mathbb{P}$ Can we find the relationship between the rank and the eigenspace of the…
Saket
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