System Of Semismooth Equations Research Articles

Projection on the intersection of convex sets

In this paper, we give a solution of the problem of projecting a point onto the intersection of several closed convex sets, when a projection on each individual convex set is known. The existing solution methods for this problem are sequential in nature. Here, we propose a highly parallelizable method. The key idea in our approach is the reformulation of the original problem as a system of semi-smooth equations. The benefits of the proposed reformulation are twofold: (a) a fast semi-smooth Newton iterative technique based on Clarke's generalized gradients becomes applicable and (b) the mechanics of the iterative technique is such that an almost decentralized solution method emerges. We proved that the corresponding semi-smooth Newton algorithm converges near the optimal point (quadratically). These features make the overall method attractive for distributed computing platforms, e.g. sensor networks.

A partial proximal point algorithm for nuclear norm regularized matrix least squares problems

We introduce a partial proximal point algorithm for solving nuclear norm regularized matrix least squares problems with equality and inequality constraints. The inner subproblems, reformulated as a system of semismooth equations, are solved by an inexact smoothing Newton method, which is proved to be quadratically convergent under a constraint non-degeneracy condition, together with the strong semi-smoothness property of the singular value thresholding operator. Numerical experiments on a variety of problems including those arising from low-rank approximations of transition matrices show that our algorithm is efficient and robust.

A new semismooth Newton method for NCPs based on the penalized KK function

In this paper, based on the Kanzow-Kleinmichel (KK) function, we introduce a new nonlinear complementarity problem (NCP) function: penalized KK function. We show that the function possesses desirable properties similar to those of the KK function. Based on this new NCP function, we reformulate the NCP to a system of semismooth equations. We also propose a new semismooth Levenberg–Marquardt method to solve the system of semismooth equations that employs both trust region techniques and line searches. The global and quadratic convergence properties can be established under very mild assumptions. Numerical results show the effectiveness of the proposed algorithm and also indicate that superior behaviour of the proposed new NCP function.

A new smoothing Newton method for solving constrained nonlinear equations

In this paper, a new smoothing Newton method is proposed for solving constrained nonlinear equations. We first transform the constrained nonlinear equations to a system of semismooth equations by using the so-called absolute value function of the slack variables, and then present a new smoothing Newton method for solving the semismooth equations by constructing a new smoothing approximation function. This new method is globally and quadratically convergent. It needs to solve only one system of unconstrained equations and to perform one line search at each iteration. Numerical results show that the new algorithm works quite well.

A new approach of available transfer capability incorporating wind generation

The approach of available transfer capability (denoted as ATC) incorporating wind generation has been paid very high attention since the development of wind generation. Based on the maximum function, this paper presents an ATC model. The characteristic of the new model is twofold. First, it considers wind turbines connected to power system and static security of power system simultaneously. Second, it is a system of semismooth equations and can be solved easily. By using the smoothing strategy, a smoothing Newton method is adopted for solving the proposed new ATC model. Numerical simulation results of the IEEE 30-bus and 118-bus system show that the new model and algorithm are feasible and effective. The impact of wind turbines connected to power system on ATC is also analyzed.

A globally convergent sequential linear programming algorithm for mathematical programs with linear complementarity constraints

This paper presents a sequential linear programming algorithm for computing a stationary point of a mathematical program with linear equilibrium constraints. The algorithm is based on a formulation of equilibrium constraints as a system of semi smooth equations by means of a perturbed Fischer-Burmeister functional. Using only data of the problem, we introduce a new method to update the parameter that characterizes the aforesaid perturbed functional. With the aim of avoiding the need to choose penalty parameters, we introduce a simple restoration step. Some computational results are reported.

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.

Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints

In this paper, we consider the least squares semidefinite programming with a large number of equality and inequality constraints. One difficulty in finding an efficient method for solving this problem is due to the presence of the inequality constraints. In this paper, we propose to overcome this difficulty by reformulating the problem as a system of semismooth equations with two level metric projection operators. We then design an inexact smoothing Newton method to solve the resulting semismooth system. At each iteration, we use the BiCGStab iterative solver to obtain an approximate solution to the generated smoothing Newton linear system. Our numerical experiments confirm the high efficiency of the proposed method.

A new hybrid method for nonlinear complementarity problems

In this paper, a new hybrid method is proposed for solving nonlinear complementarity problems (NCP) with P 0 function. In the new method, we combine a smoothing nonmonotone trust region method based on a conic model and line search techniques. We reformulate the NCP as a system of semismooth equations using the Fischer-Burmeister function. Using Kanzow's smooth approximation function to construct the smooth operator, we propose a smoothing nonmonotone trust region algorithm of a conic model for solving the NCP with P 0 functions. This is different from the classical trust region methods, in that when a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength is defined by a line search. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, the superlinear convergence of the algorithm is established without a strict complementarity condition.

The penalized Fischer-Burmeister SOC complementarity function

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian R 01-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

A globally convergent method based on Fischer–Burmeister operators for solving second-order cone constrained variational inequality problems

The Karush–Kuhn–Tucker system of a second-order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of Fischer–Burmeister operators over second-order cones. The Clarke generalized differential of the semismooth mapping is presented. A modified Newton method with Armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem. An illustrative example is given to show how the globally convergent method works.

The Semismooth Approach for Semi-Infinite Programming without Strict Complementarity

As a complement of our recent article [O. Stein and A. Tezel, J. Global Optim., 41 (2008), pp. 245–266], we study convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level problems. The semismooth Newton method is applied to a semismooth reformulation of the upper and lower level Karush–Kuhn–Tucker conditions by nonlinear complementarity problem functions into a semismooth system of equations. In the present paper we assume strict complementary slackness neither in the upper nor in the lower level. The auxiliary functions of the locally reduced problem then are not necessarily twice continuously differentiable. Still, we can show that a standard regularity condition for quadratic convergence of the semismooth Newton method holds under a natural assumption for semi-infinite programs.

A filter trust region method for solving semi-infinite programming problems

This paper is concerned with numerical methods for solving a semi-infinite programming problem. We first reformulate the KKT system derived from the problem into a system of semismooth equations by using the F-B NCP function. Under some conditions, a solution of the system of semismooth equations is a solution of the problem. Then we develop a filter-trust-region method for solving this system of semismooth equations. An advantage of the proposed method is that at each iteration, only a system of linear equations is solved. We prove that under standard assumptions, the iterate sequence generated by this proposed method converges globally and superlinearly. Numerical tests are also reported.

A Damped Gauss-Newton Method for the Second-Order Cone Complementarity Problem

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable ( SC 1 ) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.

Semismooth reformulation and Newton's method for the security region problem of power systems

In this paper, we investigate the reformulation of the steady state security region problem for electrical power systems. Firstly, a simple security region problem with one changeable parameter is reformulated into a system of semismooth equations, which is composed by the normal power flow equations and an additional piecewise smooth equation. Then the semismooth Newton method and the smoothing Newton method can be applied to solve the problem. Preliminary numerical results show that the method is promising. Finally, by using the smoothing technique, a more complicated security region problem, the Euclidean security region problem, is reformulated as an equality constrained optimization problem. These works provide a possibility to implement on-line calculation of the security region of electrical power systems.

Quadratic approximation and visualization of online contract-based available transfer capability region of power systems

This paper develops the available transfer capability (ATC) concept to available transfer capability region (ATCR). Based on the steady-state security, the ATC of a single-contract is reformulated to a system of semismooth equations. Along the line of the methodology, the ATC of a combination of multi-contracts is presented. The combination coefficients forms a region, which is defined as ATC region (ATCR) in this paper. According to the messages of ATCs of each single-contract, the boundary of ATCR is approximated by a set of quadratic functions via a patching technology. The approximation of the ATCR and the visualization take a novelly practical approach called offline-online calculation. IEEE 30-bus system is chosen to test the method with respect to point-to-point transfers and area-to-area transfers. The computing results are visualized through a class of graphs. The virtue of ATCR is that it can provide the global message of possible increasing transfer in the system. The characteristic of the proposed method in this paper is its behave of online calculation. Numerical tests show the effect of the new approach.

The semismooth approach for semi-infinite programming under the Reduction Ansatz

We study convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level problems where, using NCP functions, the upper and lower level Karush-Kuhn-Tucker conditions of the optimization problem are reformulated as a semismooth system of equations. Nonsmoothness is caused by a possible violation of strict complementarity slackness. We show that the standard regularity condition for convergence of the semismooth Newton method is satisfied under natural assumptions for semi-infinite programs. In fact, under the Reduction Ansatz in the lower level and strong stability in the reduced upper level problem this regularity condition is satisfied. In particular, we do not have to assume strict complementary slackness in the upper level. Numerical examples from, among others, design centering and robust optimization illustrate the performance of the method.

Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only.

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P 0 functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition.