Questions tagged [semidefinite-programming]

Semidefinite programming can be regarded as an extension of linear programming. In a semidefinite program, the goal is to optimize a linear function over the intersection of the cone of positive semidefinite matrices with some affine space.

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25 views

Is there an efficient way to do semidefinite programming with a Lyapunov equation constraint?

I am trying to numerically solve semidefinite programs of the form $$\begin{array}{ll} \underset{X,Y}{\text{minimize}} & \operatorname{tr}(AX)\\ \text{subject to} & BY + YB = X\\ & X, Y \...
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101 views

Matrix norm minimization and matrix inner product

One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, Convex Optimization, p. 170). Consider: \begin{equation}\label{eq:Lasse} \begin{aligned} &\min_{\mathbf{x}} & &...
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67 views

Matrix inequality $a X \succeq arcsin(X)$ for some $a > 0$

Let $X \in S^{n}_{+}$ be a positive semi-definite matrix with $X_{ii} = 1$ for all $i \leq n$ (thus $X$ is a correlation matrix). Since $X$ is positive semi-definite, we have $|X_{ij}| \leq 1$ for any ...
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52 views

Non-negative bivariate polynomials in a rectangle

I have been working on non-negative univariate polynomials and I found the following equivalent relationship to check if a polynomial is non-negative or not: The polynomial $g(x) = \sum_{r=0}^k y_rx^...
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27 views

Minimum operator that exceeds others (in a PSD, linear matrix inequality, sense)

Given a collection of $n$ matrices $A_i$, we could ask for the $B$ such that: $$\textrm{Minimize: }\quad \textrm{Tr}[B]$$ $$\textrm{Such that: }\forall_i\, B \succeq A_i$$ Here $\succeq$ is in the ...
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97 views

Multiplication of matrix-represented polynomials

Suppose we have a multivariate polynomial $f(\mathbf{x})$ represented by some matrix $A$, i.e. $f(\mathbf{x}) = b^T A b$, where $b = (1, x_1, x_2, \dots, x_{m-1}x_m^{n-1}, x_m^n)^T$ is a monomial ...
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45 views

Necessary and sufficient condition for non-negativeness of quartic form

A general quartic form in $n$ variables $x_i$, $i=1,\ldots,n$ looks like $$ Q(x) = \sum_{i,j,k,l} Q_{ijkl} x_i x_j x_k x_l $$ what are the (a) necessary (b) sufficient and (c) necessary and sufficient ...
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39 views

SDP relaxation vs. Monte Carlo for MaxCut: which one performs better?

the Goemans Williamson SDP relaxation of the MAXCUT problem famously gives a polynomial approximation ratio of .87856 for the MAXCUT on regular graphs. Another popular approach to obtain efficient ...
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30 views

On semidefinite relaxations with modified inner products

In maxcut we relax initially to a program with only inner products of form $\langle u_i,u_j\rangle$ where $u_i$'s are unit vectors. Refer section $1.5.1$ in http://www.ti.inf.ethz.ch/ew/courses/...
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What is the relation between different generalizations of linear programming?

Linear programming subsumed by each of Semidefinite programming (SDP) Convex programming (CXP) SOS programming (SSP) Is there any relation between each pair in the three? Are all three equivalent in ...
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58 views

Reference request for linear matrix inequality with PSD matrices

In literature, people say a spectrahedron is the following set $$\left\{x \in \mathbb{R}^d : x_1 A_1 + \cdots + x_d A_d \geq B \right\}$$ where $\geq$ is in the positive semidefinite sense. Is there a ...
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69 views

Representations in Archimedean quadratic modules

Let $\mathbb R [X] = \mathbb R [X_1,\dots,X_n]$ and $\Sigma[X] = \big\{ \, f \in \mathbb R[X] \mid \exists r \in \mathbb N, \ g_i \in \mathbb R[X] \colon f = g_1^2 + \dots + g_r^2 \,\big\}$ denote ...
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112 views

Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices

For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices: $$\Pi(A)=\mathrm{argmin}_{M\...
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56 views

Linear algebra - For symmetric matrix X $\in S^n$, prove the $a^T X a$ = $\det X \det(X_{n-1})$ , where $a_i$ = $(-1)^i M_{in} $ [closed]

Suppose we have a symmetric matrix X$\in S^n$, and $X_k$ denotes the submatrix consists of first $k$ rows and columns of X. If $\det X < 0$, but $\det X_1, ..., \det X_{n-1} > 0$. Let $a_i=(-1)^...
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1answer
335 views

Matrix Completion SDP Relaxation and Duality

I am studying the matrix Completion problem, as well as its SDP relaxation. However, I am having trouble deriving the final SDP form of the matrix completion problem. I will give some background, ...
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65 views

Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?

I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$ \inf_{x\in\mathbb{R}^n}\quad p(x), $$ where $p$ is a polynomial of even degree ...
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97 views

Generalization of Farkas' Lemma to Hermitian Matrices

I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
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2answers
182 views

Exactness of the semidefinite programming (SDP) relaxation of maximum cut (Max-Cut)

Currently, what conditions are known to be sufficient for the SDP relaxation of Max-Cut to be exact?
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85 views

Uniqueness of projection under spectral norm

I am considering $$ \min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0, $$ where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
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1answer
106 views

SDP representation of ideal polynomials for positivstellensatz refutations

If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e. $$ S = \{ x\in \mathbb{R}^n \ | \ h_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$ we can produce a ...
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1answer
118 views

Convex Hull of Outer Products of (Normalised) Nonnegative Vectors

If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by \begin{align} \...
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2answers
150 views

Subspaces of real $n \times n$ matrices of dimension $O(n)$ [closed]

The set of real $n \times n$ matrices forms a vector space over the reals. Given any set $S$ of $n \times n$ matrices, there is a basis $S' \subseteq S$ of size at most $n^2$ such that any $x \in S \...
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1answer
312 views

Advantages of hyperbolic programming over semidefinite programming?

What are the advantages of a hyperbolic program over a semi definite program? SDPs can be used to represent a wide variety of algebraic constraints. Are there constraints that can be represented in a ...
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2answers
519 views

A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
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2k views

About product of PSD matrices

In Theorem 3 in this paper, https://core.ac.uk/download/pdf/82822897.pdf, ``On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, ...
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1answer
1k views

Solving system of bilinear equations

Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system $$\left\{ x^T A_i y = b_i : i = 1,\dots,m \right\}$$ in variables $x,...
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2answers
537 views

Strict complementary slackness for semidefinite programs with strong duality

By a theorem of Goldman and Tucker it is known that if a linear program (LP) has a finite valued optimal solution, then there is an optimal primal/dual pair $(x,z)$ satisfying not only complementary ...
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2answers
252 views

Is this parametrized semidefinite program convex?

I am considering an optimization problem of the form: \begin{equation} \begin{split} f(s) &= \min_{X} \mathrm{tr}(C(s)X) \\ &\;\;\;\;\;\;\;\;\;\;\; X \ge 0, \\ &\;\;\;\;\;\;\;\;\;\;\; \...
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479 views

Can a quadratic matrix inequality constraint be convex?

I have an optimization problem with a semi-definiteness constraint: $$ N \preceq 0 $$ where the entries $N^{AB}$ of the matrix $N$ are defined through $$ N^{AB} = \sum_{i,j} x^i M_{ij}^{AB} x^j $$ The ...
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1answer
124 views

On Polynomial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers. $D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace. $A_1,\cdots,A_m$ are $n\times n$ Hermitians. We are interested in the ...
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1answer
273 views

Common lower bounds for positive semidefinite matrices

Suppose we are given positive semidefinite matrices $P_1, P_2, \dots, P_n \in \mathbb{C}^{m \times m}$. How to characterize the set $S$ of their common lower bounds $$S = \{Q \mid 0 \leq Q\leq P_i, \...
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2answers
310 views

Standard solution to semidefinite program [closed]

I have an optimization problem of the following form $$\text{minimize} \,\|Qa-b\|_2 \quad \text{ subject to } Q \succeq 0$$ where $a,b \in \mathbb{R}^n$ are given and the $n \times n$ square matrix ...
3
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1answer
249 views

Lovasz theta and circulant graphs

Let $\theta(G)$ denote Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be Lovasz upper bound for $\theta(G)$. Let $C_{2n+1}$ denote cycle graph with $2n+1$ nodes. We know following two ...
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1answer
1k views

Full rank submatrices of positive semidefinite matrix

Suppose $A$ is symmetric, positive semidefinite and all its diagonal entries are strictly positive (real coefficients - even integer if it helps). Suppose that the first $r$ rows of $A$ are linearly ...
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1answer
311 views

Maximize the determinant of Boolean combinations of positive definite matrices

I have the following optimization problem. $$\begin{array}{ll} \text{maximize} & \det \left(\sum^n_{i=1}z_i W_i \right)\\ \text{subject to} & \sum_{i=1}^n z_i = N\\ & z_i \in \{0,1\}\end{...
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63 views

Soft: Lagrange Multiplier and Intersection of Thickened Sets

Suppose I have an optimization problem of the form $$ \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x), $$ for some convex function $f$ and non-convex l.s.c. function $g$. Can we reinterpret the ...
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0answers
469 views

Why are SDP generally slow?

This is more of a conceptual question. Don't expect a highly mathematical question. Nonetheless, the questions I pose here often arise in my field (not mathematics). Usually Semidefinite Programs (...
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2answers
981 views

SDP relaxation vs LP relaxation

I have a question I hope you might be able to answer. Let's say we have an integer program for the stable set problem (or clique, not principal). \begin{equation} \begin{aligned} & \text{...
4
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111 views

Covering Number of a Positive Semidefinite Cone (Approximate the Objective of a SDP)

I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program \begin{align} \max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~...
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86 views

determine the existence of positive semi-definite matrix

Given a $d\times d$ complex matrix subspace $S$, we are asking whether there is some finite integer $n$ such that there exists a non-zero positive semi-definite matrix is orthogonal to $S^{\otimes n}$....
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1answer
877 views

Structure of a real 3x3 positive-semidefinite matrix whose eigenvalues verify the triangle inequalities

It is known that a 3 by 3 real symmetric matrix $A$ has an eigendecomposition $$ A = Q E Q^T $$ where $Q$ is an orthogonal matrix and $E$ is a diagonal matrix whose elements, $E_{11}$, $E_{22}$ and $...
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0answers
190 views

SDP based heuristics for graph coloring

This is a question about semidefinite programming heuristics for graph vertex coloring based on the Lovasz theta number such as "Approximate Graph Coloring" by Karger, Motwani and Sudan or "A ...
4
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2answers
406 views

About optimization with Renyi divergence

Can someone link me to some pedagogic example of computing the Renyi divergence between two discrete/continuous distributions? Like examples where someone has been able to obtain a neat closed form or ...
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0answers
138 views

About hyperplane separation theorem

I read in Lovasz's notes about semidefinite programs and combinatoric optimization. If $x_1A_1 + ... + x_nA_n\succ 0$ has no solution, then the linear subspace $L = x_1A_1 + ... + x_nA_n$ is ...
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2answers
3k views

Minimize Frobenius norm

My question is the following: Suppose $M$ is an $n \times n$ symmetric real matrix. I want to find an $n \times n$ symmetric real matrix X such that $|| X -M||_F$ is minimized with the constraint ...
3
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1answer
117 views

Finding a semigroup that maximizes the trace of a sum of matrices

Let $H$ be a finite semigroup containing $n$ elements from a compact group $G$. I am trying to solve $$\max_{h_i,\ h_j\ \in\ H} \operatorname{tr} \sum_{i,\ j\ \leq\ n} \rho(h_i^{-1} h_j)(A_j A_i^T)$$...
6
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1answer
268 views

when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...
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1answer
294 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq i,...
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1answer
774 views

How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable? In this context PSD means symmetric PSD.
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0answers
201 views

Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...