DC Semidefinite Programming and Cone Constrained DC Optimization: Theory and Local Search Methods
M. V. Dolgopolik
备注：
In the second version of the paper, typos and a number of small mistakes were corrected and a section containing results of numerical experiments was added
发表信息：
Computational Optimization and Applications 82:3 (2022) 649-671 and 85:3 (2023) 993-1031
In this paper, we study possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear and nonsmooth cone constrained optimization problems.
In the first part of the paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems.
The second part of the paper is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian.
The second-order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity. Second-order cone formulations can also be solved more efficiently than semidefinite programming in general. We consider the following question, posed by Lewis and Glineur, Parrilo, Saunderson: is it possible to express the general positive semidefinite cone using second-order cones? We provide a negative answer to this question and show that the 3x3 positive semidefinite cone does not admit any second-order cone representation. Our proof relies on exhibitin…
We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in $k,m$, and $n$) of the hyperbolicity cones associated with $k$th directional derivatives of polynomials of the form $p(x) = \det(\s…
We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.
We propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant \emph{factor width}. This adaptation makes the proposed method more suitable for parallelization than the standard IPM. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas …
Superlinear and quadratic convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programming problems
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The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to faci…
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If $X$ is an $n\times n$ symmetric matrix, then the directional derivative of $X \mapsto \det(X)$ in the direction $I$ is the elementary symmetric polynomial of degree $n-1$ in the eigenvalues of $X$. This is a polynomial in the entries of $X$ with the property that it is hyperbolic with respect to the direction $I$. The corresponding hyperbolicity cone is a relaxation of the positive semidefinite (PSD) cone known as the first derivative relaxation (or Renegar derivative) of the PSD cone. A spectrahedal cone is a convex cone that has a representation as the intersection of a subspace with the …
The robust truss topology optimization against the uncertain static external load can be formulated as mixed-integer semidefinite programming. Although a global optimal solution can be computed with a branch-and-bound method, it is very time-consuming. This paper presents an alternative formulation, semidefinite programming with complementarity constraints, and proposes an efficient heuristic. The proposed method is based upon the convex-concave procedure for DC (difference-of-convex) programming. It is shown that the method can often find a practically reasonable truss design within the compu…
Binary Integer Programming (BIP) problems are of interest due in part to the difficulty they pose and because of their various applications, including those in graph theory, combinatorial optimization and network optimization. In this note, we explicitly state the Lovasz-Schrijver Semidefinite Programming (SDP) relaxation (in primal-standard form) for a BIP problem, a relaxation that yields a tighter upper-bound than the canonical Linear Programming relaxation.
We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expa…
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The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce the first-order and second-order T-derivatives for the multi-vector real-valued function with the tensor T-product; and inspired by an equivalent characterization of a twice continuously T-differentiable multi-vector real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analo…
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Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported.
We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to …
We develop a natural variant of Dikin's affine-scaling method, first for semidefinite programming and then for hyperbolic programming in general. We match the best complexity bounds known for interior-point methods. All previous polynomial-time affine-scaling algorithms have been for conic optimization problems in which the underlying cone is symmetric. Hyperbolicity cones, however, need not be symmetric. Our algorithm is the first polynomial-time affine-scaling method not relying on symmetry.
It has been shown that the maximum stable set problem in some infinite graphs, and the kissing number problem in particular, reduces to a minimization problem over the cone of copositive kernels. Optimizing over this infinite dimensional cone is not tractable, and approximations of this cone have been hardly considered in literature. We propose two convergent hierarchies of subsets of copositive kernels, in terms of non-negative and positive definite kernels. We use these hierarchies and representation theorems for invariant positive definite kernels on the sphere to construct new SDP-based bo…
The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems (LPs) with $O(n)$ variables and $O(n^2)$ constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this paper, they investigate the properties of the subcones in more detail and explore larger subcones of the positive semidefinite plus nonnegative cone whose satisfaction of the requirements for membership can be detected…
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The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint, and then conv…
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 is infeasible if and only if -1 lies in the quadratic module associated to it. We also present a new exact duality theory for semidefinite programming, motivated by the real radic…
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We compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions, in terms of integrals related to Mehta's integral. Several applications for the probabilistic analysis of semidefinite programming are given.
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Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that already in the semidefinite case (even when strong duality holds) "hard" problems are known. We shade a light on a rather surprising restriction in the non-commutative world (of semidefinite programming), namely "commutative" paths and propose a new family of solvers that is able to use the full richness of "non-commutative" search directions: (primal) feasible-interior-point methods. Beside a detaile…
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In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of…
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In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite p…
In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primal-dual hybrid gradient method, allows the presence of linear inequalities without the need of adding extra slack variables and avoids solving a linear system at each iteration. More importantly, it does simultaneously compute the dual variables asso…
The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, which are based on the Approximate-Karush-Kuhn-Tucker necessary optimality condition and on the application of the reduction approach. Our definitions are strictly weaker than Robins…
We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithm includes, for instance, general forms of augmented Lagrangian, sequential quadratic programming, and interior point methods. We also compare these new conditions…
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In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of "no-gap" second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared…
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The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming (NLSDP) is established by assuming the second-order sufficient condition (SOSC) and the …
In this paper, we develop semidefinite programming (SDP) models aimed at solving optimal power flow (OPF) problems in distribution systems. We propose two models: the symmetrical SDP model which modifies the existing BFM-SDP model. Then based on the symmetrical SDP model, we develop a voltage regulation model that solves OPF problems with binding voltage constraints. To evaluate the accuracy of our proposed OPF models, we rely on OpenDSS, a power flow solver, to generate power flow solutions as the benchmarks. Comprehensive case studies are conducted showing our SDP models have better numerica…
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We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be transformed into a variable on a smooth manifold via matrix factorization. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semi…
We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent primal-dual pair over a lower-dimensional symmetric cone---namely, the cone-of-squares of a Jordan subalgebra of symmetric matrices. We present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra. We also show that minimizing rank optimizes the direct-sum decomposition of the algebra into simple ideals, y…
