Convex Semialgebraic Set Research Articles

Spectrahedral shadows and completely positive maps on real closed fields

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application we prove that the cone of copositive matrices of size n\geq5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model-theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital \mathbb{R} -linear completely positive map R\to R on a real closed field extension R of \mathbb{R} .

Hausdorff distance between convex semialgebraic sets

Hausdorff distance between convex semialgebraic sets

Families of faces and the normal cycle of a convex semi-algebraic set

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a patch—a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.

Design and analysis of a synthetic prediction market using dynamic convex sets

We present a synthetic prediction market whose agent purchase logic is defined using a sigmoid transformation of a convex semi-algebraic set defined in feature space. Asset prices are determined by a logarithmic scoring market rule. Time varying asset prices affect the structure of the semi-algebraic sets leading to time-varying agent purchase rules. We show that under certain assumptions on the underlying geometry, the resulting synthetic prediction market can be used to arbitrarily closely approximate a binary function defined on a set of input data. We also provide sufficient conditions for market convergence and show that under certain instances markets can exhibit limit cycles in asset spot price. We provide an evolutionary algorithm for training agent parameters to allow a market to model the distribution of a given data set and illustrate the market approximation using three open source data sets. Results are compared to standard machine learning methods.

Multi-objective convex polynomial optimization and semidefinite programming relaxations

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem \(({\mathrm{P}}_{z})\) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem \((P_{z})\), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem \(({\mathrm{P}}_{z})\) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

Convex Algebraic Geometry of Curvature Operators

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\geq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\geq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.

Second-Order Cone Representation for Convex Sets in the Plane

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 ${sxdeg}(K)$ 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 ${sxdeg}(K)$ can be seen as a measure for the complexity of performing semidefinite programs over the set $K$. We give several equivalent characterizations of ${sxdeg}(K)$ and use them to prove our main result: ${sxdeg}(K)\le2$ holds for any closed convex semialgebraic set $K\subseteq\mathbb{R}^2$. In other words, such $K$ can be represented using the second-order cone.

Voronoi cells of varieties

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.

On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most $d$. It defines a spectrahedron that projects down to a convex semialgebraic set containing $S$. In the best case, the projection equals the convex hull of $S$. We show that this is very often the case for sufficiently high $d$ if $S$ is compact and "bulges outwards" on the boundary of its convex hull. Now let additionally a polynomial objective function $f$ be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree $d$ is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if $S$ is compact and $d$ exceeds some bound that depends on the description of $S$ and certain characteristicae of $f$ like the mutual distance of its global minimizers on $S$.

The tropical analogue of the Helton–Nie conjecture is true

Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods.

Spectrahedral Shadows

We show that there are many (compact) convex semialgebraic sets in euclidean space that are not spectrahedral shadows. This gives a negative answer to a question by Nemirovski, resp. it shows that the Helton--Nie conjecture is false.

Exact Algorithms for Linear Matrix Inequalities

Let $A(x) = A_0+x_1A_1+\cdots+x_nA_n$ be a linear matrix, or pencil, generated by given symmetric matrices $A_0,A_1,\ldots,A_n$ of size $m$ with rational entries. The set of real vectors $x$ such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree $d$ of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu...

Matrix convex hulls of free semialgebraic sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.

Algebraic boundaries of convex semi-algebraic sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.

Noncommutative polynomials nonnegative on a variety intersect a convex set

This paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal.

Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness

We show that the Lasserre hierarchy of semidefinite programming (SDP) relaxations with a slightly extended quadratic module for convex polynomial optimization problems always converges asymptotically even in the case of non-compact semi-algebraic feasible sets. We then prove that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point guarantees the finite convergence of the hierarchy. We do this by establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets.

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables, let PL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of g-tuples of symmetric n×n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on PL if and only if it has a weighted sum of squares representation with optimal degree bounds: p=s∗s+∑jfinitefj∗Lfj, where s,fj are matrices of noncommutative polynomials of degree no greater than deg(p)2. This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s,fj and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.

Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions

Let $\mathcal{P}=\{h_1,\dots,h_s\}\subset\mathbb{Z}[Y_1,\dots,Y_k]$, $D\geq\deg(h_i)$ for $1\leq i\leq s$, $\sigma$ bounding the bit length of the coefficients of the $h_i$'s, and let $\Phi$ be a quantifier-free $\mathcal{P}$-formula defining a convex semialgebraic set. We design an algorithm returning a rational point in $\mathcal{S}$ if and only if $\mathcal{S}\cap\mathbb{Q}\neq\emptyset$. It requires $\sigma^{\mathrm{O}(1)}D^{\mathrm{O}(k^3)}$ bit operations. If a rational point is outputted, its coordinates have bit length dominated by $\sigma D^{\mathrm{O}(k^3)}$. Using this result, we obtain a procedure for deciding whether a polynomial $f\in\mathbb{Z}[X_1,\dots,X_n]$ is a sum of squares of polynomials in $\mathbb{Q}[X_1,\dots,X_n]$. Denote by $d$ the degree of $f$, $\tau$ the maximum bit length of the coefficients in $f$, $D={n+d\choose n}$, and $k\leq D(D+1)-{n+2d\choose n}$. This procedure requires $\tau^{\mathrm{O}(1)}D^{\mathrm{O}(k^3)}$ bit operations, and the coefficients of the outputted polynomials have bit length dominated by $\tau D^{\mathrm{O}(k^3)}$.

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

A set $S\subseteq\mathbb{R}^n$ is called semidefinite programming (SDP) representable or semidefinite representable if S equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to constructing SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\dots,W_m$, we give an explicit construction of an SDP representation for $\mathrm{conv}(\cup_{k=1}^mW_k)$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary $\partial S$ is nonsingular and positively curved, while necessary is: $\partial S$ has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on $\partial S$ and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity at those points on $\partial S$ where they vanish. The gaps between them are $\partial S$ having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is $\partial_c T$, the maximum subset of $\partial T$ contained in $\partial\mathrm{conv}(T)$. We prove sufficient for SDP representability: $\partial_c T$ is nonsingular and positively curved, and necessary is: $\partial_c T$ has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.

Complexity of integer quasiconvex polynomial optimization

We study a particular case of integer polynomial optimization: Minimize a polynomial F ^ on the set of integer points described by an inequality system F 1 ⩽ 0 , … , F s ⩽ 0 , where F ^ , F 1 , … , F s are quasiconvex polynomials in n variables with integer coefficients. We design an algorithm solving this problem that belongs to the time-complexity class O ( s ) · l O ( 1 ) · d O ( n ) · 2 O ( n 3 ) , where d ⩾ 2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of Lenstra's algorithm [Math. Oper. Res. 8 (1983) 538–548] in integer linear optimization. In the considered case, our complexity-result improves the algorithm given by Khachiyan and Porkolab [Discrete Comput. Geom. 23 (2000) 207–224] for integer optimization on convex semialgebraic sets.