The tensor complementarity problem $(\q, \mathcal{A})$ is to
$$\mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0.$$ We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem $(\q, \mathcal{A})$ has a unique solution for $\q>\0$ ($\q\geq\0$), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem $(\q, \mathcal{A})$ has a solution for all $\q \in \mathbb{R}^n$.
In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the involved functions are special polynomials defined by a tensor. Semipositive and strictly semipositive tensors play an important role in the study of the tensor complementarity problem. The article considers some important properties of semipositive tensor. We establish invariance property of semipositive tensor. We prove necessary and sufficient conditions for a t…
This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenv…
It is worth knowing that a particular tensor class belongs to $P$-tensor which ensures the compactness to solve tensor complementarity problem (TCP). In this study, we propose a new class of tensor, Nekrasov $Z$ tensor, in the context of the tensor complementarity problem. We show that the class of $P$-tensor contains the class of even ordered Nekrasov $Z$ tensors with positive diagonal elements. In this context, we propose a procedure by which a Nekrasov $Z$ tensor can be transformed into a tensor which is diagonally dominant. Keywords: Diagonally dominant tensor, Nekrasov tensors, Nonsingula…
In multilinear algebra, some special classes of matrices are extended to higher order structured tensors. The local $w$-uniqueness solution to the linear complementarity problem can be identified by the column competent matrix. Motivated by this $w$-uniqueness property, we introduce column competent tensor in the context of tensor complementarity problem. We consider some important properties. In the theory of linear complementarity problem, column competent matrices are introduced to study local $w$-uniqueness property of LCP solution. We present the inheritance property and invariance proper…
The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved that a tensor is an S-tensor if and only if the tensor complementarity problem is feasible, and each Q-tensor is an S-tensor. Furthermore, the boundedness of solution set of the tensor complementarity problem is equivalent to the uniqueness of solution for such a problem with zero vector. For the tensor complementarity problem with a strictly semi-positive te…
In this article we introduce column adequate tensor in the context of tensor complementarity problem and consider some important properties. The tensor complementarity problem is a class of nonlinear complematarity problems with the involved function being defined by a tensor. We establish the inheritance property and invariant property of column adequate tensors. We show that TCP$(q,\mathcal{A})$ has $ω$-unique solution under some assumptions.
The main purpose of this paper is devoted to an introduction of the stochastic tensor complementarity problem. We consider the expected residual minimization formulation of the stochastic tensor complementarity problem. We show that the solution set of the expected residual minimization problem is nonempty and bounded, if the associated tensor is an $R_0$ tensor. We also prove that the associated tensor being a stochastic $R_0$ tensor is a necessary and sufficient condition for the solution set of the expected residual minimization problem to be nonempty and bounded.
The paper aims to propose a suitable method in finding the solution of tensor complementarity problem. The tensor complementarity problem is a subclass of nonlinear complementarity problems for which the involved function is defined by a tensor. We propose a new homotopy function with smooth and bounded homotopy path to obtain solution of the tensor complementarity problem under some conditions. A homotopy continuation method is developed based on the proposed homotopy function. Several numerical examples are provided to show the effectiveness of the proposed homotopy continuation method.
Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular tensor, which is relevant to the tensor complementarity problem. We show that this class of tensors is a wide class of tensors which includes many important structured tensors as its special cases. By constructing two examples, we demonstrate that an exceptionally regular tensor can be, but not always, an $R$-tensor. We also show that within the class of the semi-positive tensors, the class of exceptional…
We introduce a Kojima-Megiddo-Mizuno type continuation method for solving tensor complementarity problems. We show that there exists a bounded continuation trajectory when the tensor is strictly semi-positive and any limit point tracing the trajectory gives a solution of the tensor complementarity problem. Moreover, when the tensor is strong strictly semi-positive, tracing the trajectory will converge to the unique solution. Some numerical results are given to illustrate the effectiveness of the method.
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.
Our purpose is to investigate the local boundedness, the upper semicontinuity, and the stability of the solution map of tensor complementarity problems. To do this, we focus on the set of R$_0$--tensors and show that this set plays an important role in the investigation. Furthermore, by using a technique in semi-algebraic geometry, we obtain some results on the finite-valuedness and the lower semicontinuity of the solution map.
In view of the KS-tensor complementarity problem, the sparse solution of this problem is studied. Due to the nonconvexity and noncontinuity of the l_0-norm, it is a NP hard problem to find the sparse solution of the KS-tensor complementarity problem. In order to solve this problem, we transform it into a polynomial programming problem with constraints. Then we use the sequential quadratic programming (SQP) algorithm to solve this transformed problem. Numerical results show that the SQP algorithm can find the sparse solutions of the KS-tensor complementarity problem effectively.
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…
This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor $\mathcal{A}$ such that the tensor complementarity problem $(\q, \mathcal{A})$: $$\mbox{ finding } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0, $$ has a solution for each vector $\q \in \mathbb{R}^n$. Several subclasses of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors and semi-positive R$_0$-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its principal diagonal ent…
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, 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 …
In this paper, we prove that all H$^+$(Z$^+$)-eigenvalues of each principal sub-tensor of a strictly semi-positive tensor are positive. We define two new constants associated with H$^+$(Z$^+$)eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too. Furthermore, we present global error bound analysis for a class of the nonlinear …
In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A ne…
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…
Recently, the tensor complementarity problem (TCP for short) has been investigated in the literature. An important question involving the property of global uniqueness and solvability (GUS-property) for a class of TCPs was proposed by Song and Qi in their paper "Properties of Some Classes of Structured Tensors". In the present paper, we give an answer to this question by constructing two counter-examples. We also show that the solution set of this class of TCPs is nonempty and compact. In particular, we introduce a class of related structured tensors, and show that the corresponding TCP has th…
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…
In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound of cone eigenvalues of tensors. Moreover, some new results concerning the bounds of number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not l…
We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each iteration, only a lower dimensional system of linear equation needs to be solved. The coefficient matrices of the lower dimensional linear systems are independent of the iteration after finitely many iterations. We show that starting from zero or some nonnegative point, the method generates a sequence of iterates that converges to a solution of the problem …
Properties of solutions of the tensor complementarity problem (TCP) for structured tensors have been investigated in recent literature. In this paper, we make further contributions on this problem. Specifically, we first derive solution existence theorems for TCPs on general cones from the results studied in the nonlinear complementarity problem literature. An interesting byproduct is that conditions (e.g., strict copositivity) of solution existence results for TCPs on the nonnegative cone can be reduced to copositivity, which, to the best of our knowledge, is the weakest requirement in the cu…
In this paper, we establish a new approach to solve the tensor complementarity problem (TCP). A mixed integer programming model is given and the TCP is solved by solving the model. The TCP is shown to be formulated as an equivalent mixed integer feasibility problem. Based on the reformulation, some conditions are obtained to guarantee the solvability of the TCP. Specially, a sufficient condition is given for TCP without solutions. A necessary and sufficient condition is given for existence of solutions. We also give a concrete bound for the solution set of the TCP with positive definite tensor…
In this paper, we consider the second-order cone tensor eigenvalue complementarity problem (SOCTEiCP) and present three different reformulations to the model under consideration. Specifically, for the general SOCTEiCP, we first show its equivalence to a particular variational inequality under reasonable conditions. A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem. Then, for the symmetric and sub-symmetric SOCTEiCPs, we reformulate them as appropriate nonlinear programming problems, which are extremely beneficial for de…
This paper looks at the tensor eigenvalue complementarity problem (TEiCP) which arises from the stability analysis of finite dimensional mechanical systems and is closely related to the optimality conditions for polynomial optimization. We investigate two monotone ascent spectral projected gradient (SPG) methods for TEiCP. We also present a shifted scaling-and-projection algorithm (SPA), which is a great improvement of the original SPA method proposed by Ling, He and Qi [Comput. Optim. Appl., DOI 10.1007/s10589-015-9767-z]. Numerical comparisons with some existed gradient methods in the litera…
In this paper, we consider the {\it generalized polynomial complementarity problem} (GPCP), which covers the recently introduced {\it polynomial complementarity problem} (PCP) and the well studied {\it tensor complementarity problem} (TCP) as special cases. By exploiting the structure of tensors, we first show that the solution set of GPCPs is nonempty and compact when a pair of leading tensors is cone {\bf ER}. Then, we study some topological properties of the solution set of GPCPs under the condition that the leading tensor pair is cone ${\bf R}_0$. Finally, we study a notable global Lipschi…
In this paper, we mainly focus on the existence and uniqueness of the vertical tensor complementarity problem. Firstly, combining the generalized-order linear complementarity problem with the tensor complementarity problem, the vertical tensor complementarity problem is introduced. Secondly, we define some sets of special tensors, and illustrate the inclusion relationships. Finally, we show that the solution set of the vertical tensor complementarity problem is bounded under certain conditions, and some sufficient conditions for the existence and uniqueness of the solution of the vertical tens…
Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved $\ell_0$ norm. In this paper, a special type of tensor complementarity problems with $Z$-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particu…
In this article, we introduce the class $B$-Nekrasov tensor in the context of tensor complementarity problem. We study some tensor theoretic properties. We show that the class of B-Nekrasov tensor contains the class of Nekrasov $Z$-tensor with positive diagonal entries. We present a necessary and sufficient condition for a $B$-Nekrasov tensor. We show that the class of $P$-tensor contains the class of $B$-Nekrasov tensor. Keywords: Tensor complementarity problem, $P$-tensors, Nekrasov tensors, Nekrasov $Z$-tensor, $B$-Nekrasov tensors, Nonsingular $H$-tensor, Diagonally dominant tensor.
In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is globally and Q-linearly convergent to a solution. As an application, the obtained results are used to study the linear second order cone complementarity problem, with special emphasis on the particular case of positive definite matrices. Moreover, some computational experiments designed to investigate the practical viability of the method are presented.
Augmented Lagrangian methods for nonlinear semidefinite programming with complementarity constraints
Federal University of São Paulo（圣保罗联邦大学）；State University of Maringá（马林加州立大学）；Federal University of Paraná（巴拉那联邦大学）；Instituto Federal de Santa Catarina (IFSC)（圣卡塔琳娜联邦学院）
Complementarity problems often permit distinct solutions, a fact of major significance in optimization, game theory and other fields. In this paper, we develop a numerical technique for computing multiple isolated solutions of complementarity problems, starting from the same initial guess. This technique, called deflation, is applied in conjunction with existing algorithms that reformulate the complementarity problem as the rootfinding problem of a semismooth residual. After one solution is found, the idea of deflation is to apply operators to the arguments of the corresponding semismooth refo…
In this article, we introduce the class of semimonotone star ($E_0^s$) matrices. We establish the importance of the class of $E_0^s$-matrices in the context of complementarity theory. We show that the principal pivot transform of $E_0^s$-matrix is not necessarily $E_0^s$ in general. However, we prove that $\tilde{E_0^s}$-matrices, a subclass of the $E_0^s$-matrices with some additional conditions, is in $E_0^f$ by showing this class is in $P_0.$ We prove that LCP$(q, A)$ can be processable by Lemke's algorithm if $A\in \tilde{E_0^s}\cap P_0.$ We find some conditions for which the solution set …
One of the central problems in the theory of linear complementarity problems (LCPs) is to study the class of $Q$-matrices since it characterizes the solvability of LCP. Recently, the concept of $Q$-matrix has been extended to the case of tensor, called $Q$-tensor, which characterizes the solvability of the corresponding tensor complementarity problem -- a generalization of LCP; and some basic results related to $Q$-tensors have been obtained in the literature. In this paper, we extend two famous results related to $Q$-matrices to the tensor space, i.e., we show that within the class of strong …
In this paper, the generalized eigenvalue complementarity problem for tensors (GEiCP-T) is addressed, which arises from the stability analysis of finite dimensional mechanical systems and find applications in differential dynamical systems. The general properties of the (GEiCP-T) have been studied. We establish its relationship with the generalized tensor eigenvalue problem. It follows that if exist, the number of $λ$-solutions can be bounded. We also give some sufficient conditions for the existence of the solution. In particular, there exists a unique solution of the (EiCP-T) (i.e., $J=[n]$)…
In this paper, we first introduce R0-W and SSM-W properties for the set of matrices which is a generalization of R0 and the strictly semimonotone matrix. We then prove some existence results for the extended horizontal linear complementarity problem when the involved matrices have these properties. With an additional condition on the set of matrices, we prove that the SSM-W property is equivalent to the unique solution for the corresponding extended horizontal linear complementarity problems. Finally, we give a necessary and sufficient condition for the connectedness of the solution set of the…
In this paper, we introduce a new kernel function which differs from previous functions, and play an important role for generating a new design of primal-dual interior point algorithms for semidefinite linear complementarity problem. Its properties, allow us a great simplicity for the analysis of interior-point method, therefore the complexity of large-update primal-dual interior point is the best so far. Numerical tests have shown that the use of this function gave a big improvement in the results concerning the time and the number of iterations. so is well promising and perform well enough i…
The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a DNN relaxation of the POP, and then solves the COP by applying the bisection and projection (BP) method. The COP is expressed with a linear objective function and constraints described as a single hyperplane and two cones, which a…
This paper presents a fast first-order method for solving the quasi-static contact problem with the Coulomb friction. It is known that this problem can be formulated as a second-order cone linear complementarity problem, for which regularized or semi-smooth Newton methods are widely used. As an alternative approach, this paper develops a method based on an accelerated primal-dual algorithm. The proposed method is easy to implement, as most of computation consists of additions and multiplications of vectors and matrices. Numerical experiments demonstrate that this method outperforms a regulariz…
In this paper, a sub-optimal boundary control strategy for a free boundary problem is investigated. The model is described by a non-smooth convection-diffusion equation. The control problem is addressed by an instantaneous strategy based on the characteristics method. The resulting time independent control problems are formulated as function space optimization problems with complementarity constraints. At each time step, the existence of an optimal solution is proved and first-order optimality conditions with regular Lagrange multipliers are derived for a penalized-regularized version. The per…
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…
