The algebraic degree of semidefinite programming

Bulletin of the London Mathematical Society, 2009

In this note, we use a natural desingularization of the conormal variety of the variety of (n × n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.

Collectanea Mathematica, 2022

In this article, we show that the algebraic degree in semidefinite programming can be expressed in terms of the coefficient of a certain monomial in a doubly symmetric polynomial. This characterization of the algebraic degree allows us to use the theory of symmetric polynomials to obtain many interesting results of Nie, Ranestad and Sturmfels in a simpler way.

Kodai Mathematical Journal, 2016

In this paper we use the Bott residue formula in equivariant cohomology to show a formula for the algebraic degree in semidefinite programming.

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 .

Japan Journal of Industrial and Applied Mathematics, 2013

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given.

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...

Mathematics of Operations Research, 2013

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.

Mathematics of Operations Research, 1995

We consider optimization problems of the following type: [Formula: see text] Here, tr(·) denotes the trace operator, C and X are symmetric n × n matrices, B is a symmetric m × m matrix and A(·) denotes a linear operator. Such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max-min eigenvalue problems and with new bounds for integer programming. In the context of symmetric matrices, the simplest linear operators have the following form: [Formula: see text] where M is an arbitrary m × n matrix. In this paper, we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given.

In recent years semidefinite programming has become a widely used tool for designing more efficient algorithms for approximating hard combinatorial optimization problems and, more generally, polynomial optimization problems, which deal with optimizing a polynomial objective function over a basic closed semi-algebraic set. The underlying paradigm is that while testing nonnegativity of a polynomial is a hard problem, one can test efficiently whether it can be written as a sum of squares of polynomials by using semidefinite programming. In this note we sketch some of the main mathematical tools that underlie this approach and illustrate its application to some graph problems dealing with maximum cuts, stable sets and graph colouring.

2009

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide two algebraic certificate of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied, it also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie [6]. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.