Sampling Algebraic Varieties for Sum of Squares Programs

Journal of Symbolic Computation, 2012

Let f 1 , . . . , f p be in Q [X], where X = (X 1 , . . . , X n ) t , that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f ∈ Q[X] bounded below, consider the optimization problem of computing

2021

Standard interior point methods in semidefinite programming can be viewed as tracking a solution path for a homotopy defined by a system of bilinear equations. By viewing this in the context of numerical algebraic geometry, we employ techniques to handle various cases which can arise. Adaptive precision path tracking techniques can help navigate through ill-conditioned areas. When an optimizer is singular with respect to the first-order optimality conditions, endgames can be used to accurately approximate an optimizer. When the optimal value is not achieved, the solution path diverges to infinity. In this case, current software implementations truncate the tracking of such a path. However, by using projective space, such a path always has finite length so that the endpoint can be accurately approximated using endgames. Building on these numerical algebraic geometric methods, we design a new homotopy-based approach for solving semidefinite programs without having to first find an int...

Journal of Pure and Applied Algebra, 2009

Let f , g i , i = 1, . . . , l, h j , j = 1, . . . , m, be polynomials on R n and S := {x ∈ R n | g i (x) = 0, i = 1, . . . , l, h j (x) ≥ 0, j = 1, . . . , m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.

2003

A polynomial SDP (semidefinite programs) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmet- ric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the variables to be in

Management Science, 2000

W e propose a new algorithm for solving integer programming (IP) problems that is based on ideas from algebraic geometry. The method provides a natural generalization of the Farkas lemma for IP, leads to a way of performing sensitivity analysis, offers a systematic enumeration of all feasible solutions, and gives structural information of the feasible set of a given IP. We provide several examples that offer insights on the algorithm and its properties.

Optimization Methods and Software, 2015

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f * − ǫ) and from above by f * + ǫ(n + 1), where f * is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

The IMA Volumes in Mathematics and its Applications, 2008

Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operations of numerical algebraic geometry, although no one package provides all such capabilities. This paper contains an enumeration of the operations that an ideal software package in this field would allow. The current and upcoming capabilities of Bertini, the most recently released package in this field, are also described.

Mathematical Programming, 2010

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

SIAM Journal on Optimization, 2009

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP) and p-th order cone programming (POCP), in analogy to the algebraic degree of semidefinite programming .

EURO Journal on Computational Optimization, 2015

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P ) : f * = min{ f (x) : x ∈ K } on a compact basic semi-algebraic set K ⊂ R n . This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine's positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) In contrast to the standard SOShierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems. Finally (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.