The linear complementarity problem (LCP) is a general set membership problem that includes quadratic cone programming as a special case. In this work we consider a homogeneous embedding of the LCP, which encodes both the optimality conditions of the original LCP as well as certificates of infeasibility. The resulting problem can be expressed as a monotone inclusion problem involving the sum of two maximal monotone operators, to which we apply Douglas-Rachford splitting. The resulting algorithm has almost identical per-iteration cost as applying the splitting to the LCP directly. Specifically, each iteration requires solving a linear system and projecting onto a convex cone.
The resulting algorithm has no explicit hyper-parameters and is able to return a primal-dual solution should one exist or a certificate of infeasibility otherwise. If a sequence of related problems are solved it can easily be warm-started and make use of factorization caching of the linear system. We demonstrate on a range of public and synthetic datasets that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the LCP, and in cases of infeasibility our approach can be significantly faster than alternative approaches based on diverging iterates. The algorithm we describe has been implemented in C and is available open-source as SCS-3.0.
We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem (HCCP). Employing the $T$-algebraic characterization of homogeneous cones, we generalize the $P, P_0, R_0$ properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of HCCP. We prove that if a continuous function has either the order-$P_0$ and $R_0$, or the $P_0$ and $R_0$ properties then all the associated HCCPs have solutions. In particular, if a continuous function has the trace-$P$ property then the associated HCCP has a unique so…
For a linear complementarity problem, we present a relaxaiton accelerated two-sweep matrix splitting iteration method. The convergence analysis illustrates that the proposed method converges to the exact solution of the linear complementarity problem when the system matrix is an $H_+$-matrix and the convergence conditions are given. Numerical experiments show that the proposed method is more efficient than the existing ones.
Given a polynomial map f on the Euclidean n-space and a vector q, the polynomial complementarity problem, PCP(f,q), is the nonlinear complementarity problem of finding a nonnegative vector x such that y=f(x)+q is nonnegative and orthogonal to x. It is called a tensor complementarity problem if the polynomial map is homogeneous. In this paper, we establish results connecting the polynomial complementarity problem PCP(f,q) and the tensor complementarity problem PCP(f*,0), where f* is the leading term in the decomposition of f as a sum of homogeneous polynomial maps. We show, for example, that PC…
Consider a homogeneous multifold convex conic system $$ Ax = 0, \; x\in K_1\times...\times K_r $$ and its alternative system $$ A\transp y \in K_1^*\times...\times K_r^*, $$ where $K_1,..., K_r$ are regular closed convex cones. We show that there is canonical partition of the index set ${1,...,r}$ determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming.
We propose a new error bound for the solution of tensor complementarity problem TCP$(q, \mathcal{A})$ given that $\mathcal{A}$ is a $P$-tensor and $q$ is a real vector. We show that the proposed error bound is sharper than the earlier version of error bound available in the literature. We establish absolute and relative error bound for TCP$(q, \mathcal{A})$ where $\mathcal{A}$ is an even order positive diagonal tensor. Keywords: Tensor complementarity problem, $P$-tensor, global error bound, positively homogeneous operator.
Tensors are multidimensional analogs of matrices. In this paper, based on degree-theoretic ideas, we study homogeneous nonlinear complementarity problems induced by tensors. By specializing this to $Z$-tensors (which are tensors with non-positive off-diagonal entries), we describe various equivalent conditions for a $Z$-tensor to have the global solvability property. We show by an example that the global solvability need not imply unique solvability and provide a sufficient and easily checkable condition for unique solvability.
In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem (QEiCP) for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically…
In this paper, we consider a class of $n$-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the mult…
In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose, firstly, we present that each B tensor is strictly semi-positive and each B$_0$ tensor is semi-positive. Subsequencely, the strictly lower and upper bounds of different operator norms are given for two positively homogeneous operators defined by B tensor. Finally, with the help of the upper bounds of different operator norms, we show the strcitly lower bound …
This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation which introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing stability of such systems i…
We introduce a first order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approa…
On the Local Linear Convergence of Operator Splitting Methods for Conic Programming
University of California San Diego（加州大学圣地亚哥分校）；MIT, Sloan School of Management（麻省理工学院斯隆管理学院）；University of Chicago, Department of Statistics（芝加哥大学统计学系）
In this paper, what we concern about is the weakly homogeneous variational inequality over a finite dimensional real Hilbert space. We achieve an existence result {under} copositivity of leading term of the involved map, norm-coercivity of the natural map and several additional conditions. These conditions we used are easier to check and cross each other with those utilized in the main result established by Gowda and Sossa (Math Program 177:149-171, 2019). As a corollary, we obtain a result on the solvability of nonlinear equations with weakly homogeneous maps involved. Our result enriches the…
A One-Factorization Predictor-Corrector Long-Step Arc-Search Method and a Curvature-Amplified Variant for Semidefinite Programming with a Homogeneous Self-Dual Embedding
We propose an efficient first-order method, based on the alternating direction method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit cho…
We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection…
The last two decades witnessed the increasing of the interests on the absolute value equations (AVE) of finding $x\in\mathbb{R}^n$ such that $Ax-|x|-b=0$, where $A\in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^n$. In this paper, we pay our attention on designing efficient algorithms. To this end, we reformulate AVE to a generalized linear complementarity problem (GLCP), which, among the equivalent forms, is the most economical one in the sense that it does not increase the dimension of the variables. For solving the GLCP, we propose an inexact Douglas-Rachford splitting method which can adop…
