Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinite representable set is shown to be semidefinite representable. More general, one can remove faces of a semidefinite representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.
This paper studies a class of so-called linear semi-infinite polynomial programming (LSIPP) problems. It is a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are basic semialgebraic sets. We present a hierarchy of semidefinite programming (SDP) relaxations for LSIPP problems. Convergence rate analysis of the SDP relaxations is established based on some existing results. We show how to verify the compactness of feasible sets of LSIPP problems. In the end, we extend the SDP relaxation method to more general semi-infini…
Semidefinite programs (SDPs) -- some of the most useful and versatile optimization problems of the last few decades -- are often pathological: the optimal values of the primal and dual problems may differ and may not be attained. Such SDPs are both theoretically interesting and often impossible to solve; yet, the pathological SDPs in the literature look strikingly similar. Based on our recent work \cite{Pataki:17} we characterize pathological semidefinite systems by certain {\em excluded matrices}, which are easy to spot in all published examples. Our main tool is a normal (canonical) form of …
We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box or an ellipsoid, we provide a method to compute certified outer approximations of the reachable set. The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the …
There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadrati…
In this paper, we develop a method for computing controlled invariant sets using Semidefinite Programming. We apply our method to the controller design problem for switching affine systems with polytopic safe sets. The task is reduced to a semidefinite programming problem by enforcing an invariance relation in the dual space of the geometric problem. The paper ends with an application to safety critical model predictive control.
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polyn…
Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre's semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations for representing the intersection of the sets. Properties of the semidefinite relaxations are studied. Based on that, a semidefinite relaxation algorithm is given for solving the split feasibility problem. Under a general condition, we prove that: if the split feasibility problem is feasible, we can get a feasible point; if it is infeasible, we can obtain a ce…
Stability and stabilization for linear state feedback control systems in the presence of sensor quantization are studied. As the closed-loop system is described by a discontinuous right-hand side differential equation, Krasovskii solutions (to the closed-loop system) are considered. Sufficient conditions in the form of matrix inequalities are proposed to characterize uniform global asymptotic stability of a compact set containing the origin. Such conditions are shown to be always feasible whenever the quantization-free closed-loop system is asymptotically stable. Building on the obtained condi…
The alternating-current optimal power flow (ACOPF) is one of the best known non-convex non-linear optimisation problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-voltage rank-constrained formulation, and makes it possible to derive alternative SDP relaxations. For those, we develop a first-order method based on the parallel coordinate descent with a novel closed-form step based on roots of cubic polynomials.
This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless compressed sensing, and other applications in signal processing. Results related to the generalized Kalman-Yakubovich-Popov lemma in linear system theory provide simple, constructive proofs of the semidefinite representations of the penalty functions used in these applications. The connection leads to several extensions to gauge functions and atomic norms for sets of…
The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order $n$ - the number of vertices of the graph - that is equivalent to the currently strongest semidefinite programming relaxation obtained by using vector lifting. The reduction in the size of the matrix variable enables us to impose additional valid inequalities to the relaxation in order to further …
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and Yildiz [18], using the sum of squares cone directly in a nonsymmetric interior point algorithm. Beyond nonnegativity, more complicated polynomial constraints (in particular, generalizations of the positive semidefinite, second order and $\ell_1$-norm cones) can also be modeled through structured sum of squares programs. We take a different approach and propo…
Given integers n $\geq$ m, let Sep(n,m) be the set of separable states on the Hilbert space $\mathbb{C}^n \otimes \mathbb{C}^m$. It is well-known that for (n,m)=(3,2) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set Sep(n,m) has no semidefinite programming description of finite size. As Sep(n,m) is a semialgebraic set this provides a new counterexample to the Helton-Nie conjecture, which was recently disproved by Scheiderer…
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals $[a,b]$, $[0,\infty)$; and the strips $[a,b] \times \mathbb{R} \subset \mathbb{R}^2.$ A method for numerically computing such representations is also presented. The methodology is divided into two stages: (S1) diagonalizing the initial polynomial matrix based on the Schmüdgen's procedure \cite{Schmudgen09}; (S2) for each diagonal element of the resulting matrix,…
Eigenvalue, Quadratic Programming, and Semidefinite Programming Bounds for a Cut Minimization Problem
考虑将图的节点集划分为给定大小的$k$集，从而得到使用（去除）第$k$集得到的\emph{minimize the cut}的问题.如果得到的割具有值$0$，那么我们得到了顶点分隔符。这个问题与图的划分问题密切相关。事实上，我们使用的模型与图划分问题的模型相同，除了不同的\emph{quadratic}目标函数。我们期待在已知的和新的界限，从各种放宽这个NP-困难的问题。这包括：标准特征值界、使用邻接矩阵和拉普拉斯算子的投影特征值界、基于最近成功的二次指派问题的二次规划（QP）界的QP界、以及半定规划界。我们包括大型和\emph{huge}的问题，说明在强度和时间方面的边界的效率的数值测试。
We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, …
We prove that every pointed closed convex set in $\mathbb{R}^n$ is the intersection of all the rational closed halfspaces that contain it. This generalizes a previous result by the authors for compact convex sets.
Semidefinite programming (SDP) is the task of optimizing a linear function over the common solution set of finitely many linear matrix inequalities (LMIs). For the running time of SDP solvers, the maximal matrix size of these LMIs is usually more critical than their number. The semidefinite extension degree $\text{sxdeg}(K)$ of a convex set $K\subseteq\mathbb R^n$ is the smallest number $d$ such that $K$ is a linear image of a finite intersection $S_1\cap\dots\cap S_N$, where each $S_i$ is a spectrahedron defined by a linear matrix inequality of size $\le d$. Thus $\text{sxdeg}(K)$ can be seen…
The aim of this paper is twofold. On one hand the generalized Minkowski sets are defined and characterized. On the other hand, the Motzkin decomposable sets, along with their epigraphic versions are considered and characterized in new ways. Among them, the closed convex sets with one single minimal face, i.e. translated closed convex cones, along with their epigraphic counterparts are particularly studied.
We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $μ(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non decreasing) sequences of upper and lower bounds $(\overlineω\_d)$, $(\underlineω\_d)$, $d\in\N$, each converging to $μ(\om)$ as $d\to\infty$. For each $d$, computing $\overlineω\_d$ or $\underlineω\_d$reduces to solving a semidefinite program whose size increases with $d$. Some preliminary (small dimension) computational experiments are encouraging and illustrate the…
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation i…
We consider the semi-infinite system of polynomial inequalities of the form \[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in S\subseteq\mathbb{R}^n\}, \] where $p(x,y)$ is a real polynomial in the variables $x$ and the parameters $y$, the index set $S$ is a basic semialgebraic set in $\mathbb{R}^n$, $-p(x,y)$ is convex in $x$ for every $y\in S$. We propose a procedure to construct approximate semidefinite representations of $\mathbf{K}$. There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation se…
We study the problem of when the continuous linear image of a fixed closed convex set $X \subset\mathbb{R}^n$ is closed. Specifically, we improve the main results in the papers \cite{Borwein2009, Borwein2010} by showing that for all, except for at most a $σ$-porous set, of the linear mappings $T$ from $\mathbb{R}^n$ into $\mathbb{R}^m,$ not only $T(X)$ is closed, but there is also an open neighborhood of $T$ whose members also preserve the closedness of $X.$
