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.
95 questions
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Solving an SDP that involves a decreasing sequence of circulant and skew-circulant matrices
Question. I would like to find an analytic solution of the following semidefinite program, where $e_0 = (1, 0, \ldots, 0)^{\top}$.
$$
\begin{aligned}
y = \min &\frac{1}{n} \sum_{i,j=0}^{n-1} A[i,j]...
- 81
1
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0
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76
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Software to find certificates of positivity not involving sums of squares
I am working in a real commutative associative algebra $A$ generated by variables $x_1, \dots, x_n$, such that each $x_i$ is an hermitian $2 \times 2$ matrix variable and the product is the symmetric ...
- 5,377
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131
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Minimum eigenvalue and semidefinite cone
I have an linear matrix inequality(LMI) in the form: $G + x_1F_1 + \cdots + x_nF_n \succeq 0$, where $G$ and $F_i$ are symmetric matrices, $x_i \in \{0, 1\}$, and a matrix $A \succeq 0$ means that the ...
- 21
1
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1
answer
403
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Solving a 0-1 quadratic matrix inequality
I am working on a binary optimization problem. So far I have derived the following constraint functions.
\begin{align}
\begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
- 21
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1
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141
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Transform a matrix optimization problem into a semidefinite programming
I am working on a matrix optimization problem, and the constraints are difficult to handle.
The constraints are in the following form,
\begin{align}
\text{Given: } &b \in \mathbb{R}^n \text{ , and ...
- 21
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1
answer
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Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
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135
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Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 83
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112
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Reference request: finding entries that prevent matrix from being correlation matrix
I am currently doing some research with a quantitative finance firm and my supervisor has raised an interesting question that shows up a lot with their clients: quite often, clients will want to do ...
- 811
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Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
- 285
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Norm bound in simultaneous stability to semidefinite program
In the context of robust control, I remember hearing that the two following problems are equivalent.
Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
- 285
1
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1
answer
251
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Perturbation of positive semidefinite matrix
Consider an $n\times n$ matrix $A$ that is positive semidefinite and has rank $n-1$, so there exists exactly one eigenvector $v$ such that $Av=0$. Let now $B$ be a symmetric matrix such that $v^TBv=0$....
- 699
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Positive-definite block matrix with constant block sums
Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm \times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the ...
- 699
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Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
- 733
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Feasibility of a polynomial system of equalities and inequalities
Consider a system of the form $f_i(x) = 0$ and $g_j(x) \ge 0$ ,where $f_i,i=1,\dots,r$ and $g_j,j=1,\dots,s$ are polynomials in real unknowns $x_i,i=1,\dots,n$ with rational coefficients.
Is there a ...
- 1,483
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If $\hat{D}$ minimizes trace over all $D+ B \succeq 0$, then is $\hat{D}_{ii} \leq \sum_{j} |B_{ij}|$ for each $i$?
Let $A$ be an $n \times n$ real matrix and let $B$ be the block bipartite matrix
$$B = \begin{bmatrix} 0&A \\
A^{T}&0 \end{bmatrix}$$
Let $\hat{D}$ be a solution to the SDP that minimizes $tr(...
- 411
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172
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Minimizing $\det(D)$ for all diagonal matrices $D$ that satisfy $D+B \succeq 0$
Let $A$ be an $n \times n$ real matrix and let $B$ be the block bipartite matrix
$$B = \begin{bmatrix} 0&A \\
A^{T}&0 \end{bmatrix}$$
I came across the following optimization problem, which ...
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2
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Monotonicity of kernel matrices with respect to hyperparameters
Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
- 949
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Literature request: proving or disproving convexity of the optimal value function of semidefinite program (SDP) or convex optimization in general
Suppose I have a function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined as the following parametric optimization problem:
$$f(p) = \inf_xf_0(x) \quad \text{subject to } \quad G(x,p)\leq 0,$$
where ...
2
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1
answer
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Solving linear programming without solving linear programming
Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them.
It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
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173
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Modified quadratic assignment problem
Let $Y,Z$ be $n\times k$ matrices and assume all columns have been standardized such that their means are zero and variances 1. I seek an $n\times n$ permutation matrix $P$ such that
$$\left\Vert Y^{T}...
- 133
0
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0
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199
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Double summation of matrices as constraints in convex optimization in CVX
I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53:
\begin{align}
\text{minimize} &\qquad s\\
\text{subject to} & \...
2
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0
answers
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A variant of the elliptope relaxation
Given a p.s.d. matrix $A$, one may want to find:
$$
\max_x x^t A x \mbox{ such that } x \mbox{ has entries }1 \mbox{ or } {-1}.
$$
This hard problem has a well known relaxation based on the so called ...
- 2,966
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167
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On least-squares with positive semidefinite constraints
Given real symmetric matrix $\mathbf{R} \in \mathbb{S}^{n\times n}$ and matrices $\mathbf{X}_n, \mathbf{X}_{n-1} \in \mathbb{R}^{n \times m}$,
$$\begin{array}{ll} \underset{\mathbf{A} \in \mathbb{R}^{...
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Relaxations for the spectral norm maximization problem
Optimizing the spectral norm of some positive semidefinite matrix $A(x) \in S^{n}$, w.r.t. a list of variables $x \in \mathbb{R}^d$ and semidefinite constraints is, in general, a nonconvex problem (...
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answers
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Certificates of connectivity of basic semi-algebraic sets
Given real polynomials $p_1, \ldots, p_n \in {\mathbb R}[x_1, \ldots, x_d]$, consider the closed basic semi-algebraic set $S \subseteq {\mathbb R}^d$ given by $$S := \{x \in {\mathbb R}^d : p_i(x) \...
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3
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When does a finite metric induce a matrix norm?
If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit ...
- 4,131
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What is the convex cone generated by the pair of rank 1 matrix and its eigenvector?
I'd like to know what is the convex cone generated by $\left\{ (h h^T, h) : h \in \Bbb R^{d\times1} \right\}$. It is known that $$\mathrm{cone} \left\{h h^T : h \in \Bbb R^{d \times1} \right\} = S_+^d$...
- 118
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Three-constraint homogeneous QCQP
Consider the homogeneous quadratically constrained quadratic program,
$$\min_{u^T u =1} u^T A_1 u$$
$$\textrm{subject to}\quad u^T A_2 u = 0,\quad u^T A_3 u = 0$$
This problem is particularly studied ...
- 1,203
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Solution to dynamic program-type recursion
I have the following dynamic programming principle-type problem.
Suppose that we are given a sequence $\beta_1,\dots,\beta_n\in (0,\infty)$, some target $y\in (0,\infty)$ with $y>\sum_{t=1}^N \...
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2
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1
answer
108
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Convexity of a positive definite objective with min(x,y)-nonlinearity
I have derived an optimization objective of the form
$$
f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0
$$
where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \...
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Is there an efficient way to do semidefinite programming with a Lyapunov equation constraint?
I am trying to numerically solve semidefinite programs of the form
$$\begin{array}{ll} \underset{X,Y}{\text{minimize}} & \operatorname{tr}(AX)\\ \text{subject to} & BY + YB = X\\ & X, Y \...
- 31
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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}}
& &...
- 355
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votes
0
answers
156
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 ...
- 205
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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^...
- 131
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vote
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answers
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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 ...
- 1,203
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vote
0
answers
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views
Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 1
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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 ...
- 111
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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 ...
- 1,846
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answers
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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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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 ...
- 31
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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:
$$\Pi(A)=\mathrm{argmin}_{M\...
- 41
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Linear algebra - For symmetric matrix X $\in S^n$, prove the $a^T X a$ = $\det X \det(X_{n-1})$ , where $a_i$ = $(-1)^i M_{in} $ [closed]
Suppose we have a symmetric matrix X$\in S^n$, and $X_k$ denotes the submatrix consists of first $k$ rows and columns of X. If $\det X < 0$, but $\det X_1, ..., \det X_{n-1} > 0$. Let $a_i=(-1)^...
- 33
1
vote
1
answer
2k
views
Matrix Completion SDP Relaxation and Duality
I am studying the matrix Completion problem, as well as its SDP relaxation. However, I am having trouble deriving the final SDP form of the matrix completion problem. I will give some background, ...
1
vote
0
answers
134
views
Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?
I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy:
$$
\inf_{x\in\mathbb{R}^n}\quad p(x),
$$
where $p$ is a polynomial of even degree ...
- 617
2
votes
0
answers
180
views
Generalization of Farkas' Lemma to Hermitian Matrices
I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
6
votes
3
answers
928
views
Exactness of the semidefinite programming (SDP) relaxation of maximum cut (Max-Cut)
Currently, what conditions are known to be sufficient for the SDP relaxation of Max-Cut to be exact?
- 61
3
votes
0
answers
204
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Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
- 131
3
votes
1
answer
249
views
SDP representation of ideal polynomials for positivstellensatz refutations
If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e.
$$ S = \{ x\in \mathbb{R}^n \ | \ h_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$
we can produce a ...
- 31
4
votes
1
answer
367
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}
\...
- 949
2
votes
2
answers
206
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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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