In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems in an iterative way.
The proximal, regular and limiting normal cones to the second-order cone complementarity set play important roles in studying mathematical programs with second-order cone complementarity constraints, second-order cone programs, and the second-order cone complementarity problems. It is needed in the first-order optimality conditions for mathematical programs with second-order cone complementarity constraint, the second-order subdifferential criteria in characterizing the full stability for second-order cone programs and second-order cone complementarity problems, as well as in the characterizin…
The congruence orbit of a matrix has a natural connection with the linear complementarity problem on simplicial cones formulated for the matrix. In terms of the two approaches -- the congruence orbit and the family of all simplicial cones -- we give equivalent classification of matrices from the point of view of the complementarity theory.
In this paper, we study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We state necessary and sufficient conditions for a point to be a solution of the converted problem. We also present solution strategies for this problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally, we present some numerical examples.
The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC compl…
In this paper, we study a new generalization of the Lorentz cone, called the Monotone Extended Second Order Cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a Nonlinear Complementarity Problem on a cylinder, which is equivalent to a suitable Mixed Complementarity Problem and provide a computational example illustrating applicability of MESOC.
In this paper, we study the stochastic linear complementarity problems on extended second order cones (stochastic ESOCLCP). We first convert the problem to a stochastic mixed complementarity problem on the nonegative orthant (SMixCP). Enlightened by the idea of Chen and Lin(2011), we introduce the Conditional Value-at-risk (CVaR) method to measure the loss of complementarity in the stochastic case. A CVaR - based minimisation problem is introduced to achieve a solution which is "good enough" for the complementarity requirement of the original SMixCP. Smoothing function and sample average appro…
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.
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 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…
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…
In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We present algorithms for this problem, and exemplify it by a numerical example. Following this result, we explore the stochastic version of this linear complementarity problem. Finally, we apply complementarity problems on extended second order cones in a portfolio optimisation problem. In this application…
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…
The error bound property for a solution set defined by a set-valued mapping refers to an inequality that bounds the distance between vectors closed to a solution of the given set by a residual function. The error bound property is a Lipschitz-like/calmness property of the perturbed solution mapping, or equivalently the metric subregularity of the underlining set-valued mapping. It has been proved to be extremely useful in analyzing the convergence of many algorithms for solving optimization problems, as well as serving as a constraint qualification for optimality conditions. In this paper, we …
The MaxCut SDP is one of the most well-known semidefinite programs, and it has many favorable properties. One of its nicest geometric/duality properties is the fact that the vertices of its feasible region correspond exactly to the cuts of a graph, as proved by Laurent and Poljak in 1995. Recall that a boundary point $x$ of a convex set $C$ is called a vertex of $C$ if the normal cone of $C$ at $x$ is full-dimensional. We study how often strict complementarity holds or fails for the MaxCut SDP when a vertex of the feasible region is optimal, i.e., when the SDP relaxation is tight. While strict…
The Lyapunov rank of a proper cone $K$ in a finite dimensional real Hilbert space is defined as the dimension of the space of all Lyapunov-like transformations on $K$, or equivalently, the dimension of the Lie algebra of the automorphism group of $K$. This (rank) measures the number of linearly independent bilinear relations needed to express a complementarity system on $K$ (that arises, for example, from a linear program or a complementarity problem on the cone). Motivated by the problem of describing spectral/proper cones where the complementarity system can be expressed as a square system (…
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 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 necessary conditions and sufficient conditions are given for a linear operator to be a positive operators of an Extended Lorentz cone. Similarities and differences with the positive operators of Lorentz cones are investigated.
The Z-property of a linear map with respect to a cone is an extension of the notion of Z-matrices. In a recent paper of Orlitzky (see Corollary 6.2 in M. Orlitzky. Positive and $\mathbf{Z}$-operators on closed convex cones, Electron. J Linear Algebra, 444--458, 2018) the characterisation of cone-complementarity is given in terms of the dual of the cone of linear maps satisfying the Z-property. Therefore, it is meaningful to consider the problem of finding the dual cone of the cone of linear maps which have the Z-property with respect to a cone. This short note will solve this problem in the pa…
Proper cones with the property that the projection onto them is isotone with respect to the order they induce are called isotone projection cones. Isotone projection cones and their extensions have been used to solve complementarity problems and variational inequalities. Q-matrices are matrices with the property that all classical linear complementarity problems defined by them are solvable. This note will use the isotone projection cones to generate a large class of Q-matrices. More specifically, it will be shown that the product between a non-negative matrix with positive diagonal elements a…
This paper presents a canonical dual approach for solving nonconvex quadratic minimization problem. By using the canonical duality theory, nonconvex primal minimization problems over n-dimensional Lorentz cone can be transformed into certain canonical dual problems with only one dual variable, which can be solved by using standard convex minimization methods. Extremality conditions of these solutions are classified by the triality theory. Applications are illustrated.
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.
Solutions of a variational inequality are found by giving conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to its natural map. One of these conditions is the isotonicity of the projection onto the closed convex set in the definition of the variational inequality. If the closed convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satisfied. The obtained result is further particularized for unbounded box constrained variational inequalities. For this case a numerical e…
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson's constraint qualification and a weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence result…
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…
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 provide an elementary, geometric, and unified framework to analyze conic programs that we call the strict complementarity approach. This framework allows us to establish error bounds and quantify the sensitivity of the solution. The framework uses three classical ideas from convex geometry and linear algebra: linear regularity of convex sets, facial reduction, and orthogonal decomposition. We show how to use this framework to derive error bounds for linear programming (LP), second order cone programming (SOCP), and semidefinite programming (SDP).
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, …
A weighted complementarity problem (wCP) is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem (wHLCP) in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a …
This paper is concerned with second-order optimality conditions for the mathematical program with semidefinite cone complementarity constraints (SDCMPCC).To achieve this goal, we first provide an exact characterization on the second-order tangent set to the semidefinite cone complementarity (SDCC) set in terms of the second-order directional derivative of the projection operator onto the SDCC set, and then use the second-order tangent set to the SDCC set to derive second-order necessary and sufficient conditions for SDCMPCC under suitable subregularity constraint qualifications. An application…
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…
The extended second order cones were introduced by S. Z. Németh and G. Zhang in [S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. Németh and G. Zhang in [S.Z. Németh and G. Zhang. Positive operator…
In this paper the simplicial cone constrained convex quadratic programming problem is studied. The optimality conditions of this problem consist in a linear complementarity problem. This fact, under a suitable condition, leads to an equivalence between the simplicial cone constrained convex quadratic programming problem and the one of finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to this nonsmooth system of equations is always well defined and under a mild assumption on the simplicial cone the method generates a sequence th…
Let $C$ be a closed convex subset of a real Hilbert space containing the origin, and assume that $K$ is the homogenization cone of $C$, i.e., the smallest closed convex cone containing $C \times \{1\}$. Homogenization cones play an important role in optimization as they include, for instance, the second-order/Lorentz/"ice cream" cone. In this note, we discuss the polar cone of $K$ as well as an algorithm for finding the projection onto $K$ provided that the projection onto $C$ is available. Various examples illustrate our results.
We use the concept of barrier-based smoothing approximations introduced in [ C. B. Chua and Z. Li, A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones, SIOPT 23(2), 2010] to extend the non-interior continuation method proposed in [B. Chen and N. Xiu, A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIOPT 9(3), 1999] to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the me…
We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically …
