Monotone Complementarity Problems Research Articles

Iterative complexities of a class of homogeneous algorithms for monotone nonlinear complementarity problems over symmetric cones

ABSTRACTThis paper provides an analysis of the iterative complexities of a class of homogeneous algorithms for monotone nonlinear complementarity problems over symmetric cones. The proof of the complexity bounds requires that the nonlinear transformation satisfies an SLC. To prove the complexity bounds of the homogeneous algorithm, this paper proposes an SLC which does not depend on the scaling parameter p and is very easy to be verified. More important, it has scaled invariance. Underlying SLC, the obtained complexity bounds of the short-step algorithm, the semi-long-step algorithm and the long-step algorithm with the sx-direction match that of the homogeneous algorithms proposed by Yoshise [Homogenous algorithms for monotone complementarity problems over symmetric cones. Pac J Optim. 2008; 5(2):313–337].

Solution of Monotone Complementarity and General Convex Programming Problems Using a Modified Potential Reduction Interior Point Method

We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition (SLC) is satisfied. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation in iOptimize maintains global linear and polynomial time convergence properties, while achieving practical performance. It either successfully solves the problem, or concludes that the SLC is not satisfied. When compared with the mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. We also find that iOptimize detects infeasibility more reliably than the general nonlinear solvers Ipopt (version 3.9.2) and Knitro (version 8.0).

Interior hybrid proximal extragradient methods for the linear monotone complementarity problem

We present new infeasible path-following methods for linear monotone complementarity problems based on Auslender, Teboulle and Ben-Tiba’s log-quadratic barrier functions. The central paths associated with these barriers are always well defined and, for those problems which have a solution, convergent to a pair of complementary solutions. Starting points in these paths are easy to compute. The theoretical iteration-complexity of these new path-following methods is derived and improved by a strategy which uses relaxed hybrid proximal-extragradient steps to control the quadratic term. Encouraging preliminary numerical experiments are presented.

Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function

This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.

An inexact smoothing method for the monotone complementarity problem over symmetric cones

We present an inexact smoothing method for the monotone complementarity problem over symmetric cones (SCCP). Our algorithm needs only to solve one linear system of equation and perform one line search per iteration. Instead of solving the linear equation exactly, we only need an inexact solution with a certain degree of accuracy. It is shown that any accumulation point of generated sequence is a solution of SCCP. It is proved that the proposed algorithm is locally superlinearly/quadratically convergent under suitable conditions. The computational results show the feasibility and efficiency of our algorithm.

A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators

The purpose of this paper is to introduce a new iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi‐ϕ‐nonexpansive mappings in Banach spaces. Moreover, the strong convergence theorems of this method are established under the suitable conditions of the parameter imposed on the algorithm. Finally, we apply our results to finding a zero point of inverse‐strongly monotone operators and complementarity problems. Our results presented in this paper improve and extend the recently results by many others.

Nonnegative inverse eigenvalue problems with partial eigendata

In this paper we consider the inverse problem of constructing an n × n real nonnegative matrix A from the prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem. To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth Newton-type method for solving its equivalent nonsmooth equation. Under some mild assumptions, the global and quadratic convergence of our method is established. We also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [L. Faybusovich, Positivity, 1 (1997), pp. 331-357] depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang [Math. Oper. Res., 23 (1998), pp. 39-60], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Consequently, we show the existence of a path having the following properties: (a) The path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point of the path is a solution of the homogeneous model. (c) If the original problem is solvable, then every accumulation point of the path gives us a finite solution. (d) If the original problem is strongly infeasible, then, under the assumption of Lipschitz continuity, any accumulation point of the path gives us a finite certificate proving infeasibility.

A Jordan-algebraic approach to potential-reduction algorithms

We consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the Cartesian product of symmetric cones. A primal-dual potential reduction algorithm is described and its complexity estimates are established with the help of the Jordan-algebraic technique.

A Relative Error Tolerance for a Family of Generalized Proximal Point Methods

We propose a new kind of inexact scheme for a family of generalized proximal point methods for the monotone complementarity problem. These methods, studied by Auslender, Teboulle, and Ben-Tiba, converge under the sole assumption of existence of solutions. We prove convergence of our new scheme and discuss its implementability.

On a New Homotopy Continuation Trajectory for Nonlinear Complementarity Problems

Most known continuation methods for P0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and the properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated. For P0 complementarity problems, the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods. Particularly, the often-assumed strictly feasible condition is not required here. When applied to P* complementarity problems, the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem, and the entire trajectory converges to the (unique) least element solution provided that it exists. Moreover, for monotone complementarity problems, the whole trajectory always converges to a least 2-norm solution provided that the solution set of the problem is nonempty. The results presented in this paper can serve as a theoretical basis for constructing a new path-following algorithm for solving complementarity problems, even for the situations where the solution set is unbounded.

On Primal and Dual Infeasibility Certificates in a Homogeneous Model for Convex Optimization

Andersen and Ye [ Math. Programming, 84 (1999), pp. 375--399] suggested a homogeneous formulation and an interior-point algorithm for solution of the monotone complementarity problem (MCP). The advantage of the homogeneous formulation is that it always has a solution. Moreover, in the case in which the MCP is solvable or is (strongly) infeasible, the solution provides a certificate of optimality or infeasibility. In this note we demonstrate that if the suggested formulation is applied to the Karush--Kuhn--Tucker optimality conditions corresponding to a convex optimization problem, then an infeasibility certificate provides information about whether the primal or dual problem is infeasible given certain assumptions.

A quadratically convergent polynomial long-step algorithm for A class of nonlinear monotone complementarity problems*

Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worst-case complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper presents an algorithm which is both long-step and polynomial. In addition, the sequence generated by the algorithm, as well as the corresponding complementarity gap, converges quadratically. The proof of the polynomial complexity requires that the monotone mapping satisfies a scaled Lipschitz condition, while the quadratic rate of convergence is derived under the assumptions that the problem has a strictly complementary solution and that the Jacobian of the mapping satisfies certain regularity conditions

The Proximal Point Algorithm with Genuine Superlinear Convergence for the Monotone Complementarity Problem

In this paper, we consider a proximal point algorithm (PPA) for solving monotone nonlinear complementarity problems (NCP). PPA generates a sequence by solving subproblems that are regularizations of the original problem. It is known that PPA has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems. However, it is not always easy to solve subproblems or to check those criteria. In this paper, we adopt the generalized Newton method proposed by De Luca, Facchinei, and Kanzow to solve subproblems and adopt some NCP functions to check the criteria. Then we show that the PPA converges globally provided that the solution set of the problem is nonempty. Moreover, without assuming the local uniqueness of the solution, we show that the rate of convergence is superlinear in a genuine sense, provided that the limit point satisfies the strict complementarity condition.

Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems

We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair (x,y)∈ℝ2n satisfying y=f(x) and xiyi=0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝn to ℝn. The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features; (a) it traces a trajectory in ℝ3n which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary (not necessarily positive) point in ℝ2n in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including (not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it.

Two Interior-Point Methods for Nonlinear P *(τ)-Complementarity Problems

Two interior-point algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P*-complementarity problems. The proof of the polynomial complexity of the first method requires the problem to satisfy a scaled Lipschitz condition. When specialized to monotone complementarity problems, the results of the first method are similar to those in Ref. 1. The second method is quite different from the first in that the global convergence proof does not require the scaled Lipschitz assumption. However, at each step of this algorithm, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied.

Smooth methods of multipliers for complementarity problems

This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems.

On a homogeneous algorithm for the monotone complementarity problem

We present a generalization of a homogeneous self-dual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "big-M" parameter or two-phase method, and it generates either a solution converging towards feasibility and comple- mentarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interior-point and infeasible-starting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented.

On the analyticity properties of infeasible-interior-point paths for monotone linear complementarity problems

Infeasible-interior-point paths \(\{(x(r,\eta),y(r,\eta))\}_{r>0}\), \(0 0, y>0, x_iy_i &=& r\eta_i, & 1\le i\le n. \end{array} \leqno(LCP)_r \) If the path \(z(r,\eta)=(x(r,\eta),y(r,\eta))\) converges for \(r\downarrow 0\), then it converges to a solution of \((LCP)_0\). This paper deals with the analyticity properties of \(z(r,\eta)\) and its derivatives with respect to r near r = 0 for solvable monotone complementarity problems \((LCP)_0\). It is shown for \((LCP)_0\) with a strictly complementary solution that the path \(z(r,\eta)\), \(r\downarrow0\), has an extension to \(r=0\) which is analytic also at \(r=0\). If \((LCP)_0\) has no strictly complementary solution, then \(\hat z(\rho,\eta):=z(\rho^2,\eta)\), \(\rho=\sqrt r\), has an extension to \(\rho=0\) that is analytic at \(\rho=0\).

A Potential Reduction Newton Method for Constrained Equations

Extending our previous work [T. Wang, R. D. C. Monteiro, and J.-S. Pang, Math. Programming, 74 (1996), pp. 159--195], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A major convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strengthened convergence results are established in the context of these specializations.