Merit Function Approach Research Articles

A merit function approach for evolution strategies

• A globally convergent evolution strategy to solve general constrained optimization problems. • Quantifiable relaxable constraints are handled using a merit function approach combined with a specific restoration procedure. • The proposed approach is very competitive compared to existing derivative-free optimization solvers on a large set of problems. In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles quantifiable relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints, when present, can be treated either by using the extreme barrier function or through a projection approach. Under reasonable assumptions, the introduced extension guarantees to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Numerical experiments are carried out on a set of problems from the CUTEst collection as well as on known global optimization problems.

On merit functions for p-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.

The H-differentiability and calmness of circular cone functions

Let $$\mathcal{L}_{\theta }$$L? be the circular cone in $${\mathbb {R}}^n$$Rn which includes second-order cone as a special case. For any function f from $${\mathbb {R}}$$R to $${\mathbb {R}}$$R, one can define a corresponding vector-valued function $$f^{\mathcal{L}_{\theta }}$$fL? on $${\mathbb {R}}^n$$Rn by applying f to the spectral values of the spectral decomposition of $$x \in {\mathbb {R}}^n$$x?Rn with respect to $$\mathcal{L}_{\theta }$$L?. The main results of this paper are regarding the H-differentiability and calmness of circular cone function $$f^{\mathcal{L}_{\theta }}$$fL?. Specifically, we investigate the relations of H-differentiability and calmness between f and $$f^{\mathcal{L}_{\theta }}$$fL?. In addition, we propose a merit function approach for solving the circular cone complementarity problems under H-differentiability. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.

Geometric views of the generalized Fischer–Burmeister function and its induced merit function

In this paper, we study geometric properties of surfaces of the generalized Fischer–Burmeister function and its induced merit function. Then, a visualization is proposed to explain how the convergent behaviors are influenced by two descent directions in merit function approach. Based on the geometric properties and visualization, we have more intuitive ideas about how the convergent behavior is affected by changing parameter. Furthermore, geometric view indicates how to improve the algorithm to achieve our goal by setting proper value of the parameter in merit function approach.

Lipschitz continuity of the gradient of a one-parametric class of SOC merit functions

In this article, we show that a one-parametric class of SOC merit functions has a Lipschitz continuous gradient; and moreover, the Lipschitz constant is related to the parameter in this class of SOC merit functions. This fact will lay a building block when the merit function approach as well as the Newton-type method are employed for solving the second-order cone complementarity problem with this class of merit functions.

Practical implementation of an interior point nonmonotone line search filter method

Here we present a primal-dual interior point nonmonotone line search filter method for nonlinear programming. The filter relies on three measures, the feasibility, the centrality and the optimality presented in the optimality conditions, and considers relaxed acceptability criteria for the step size and includes a feasibility restoration phase. Evaluation of the method has, until now, been made on small problems and a comparison is provided with a merit function approach.

A regularization method for the second-order cone complementarity problem with the Cartesian [formula omitted]-property

We consider the Tikhonov regularization method for the second-order cone complementarity problem (SOCCP) with the Cartesian P 0 -property. We show that many results of the regularization method for the P 0 -nonlinear complementarity problem still hold for this important class of nonmonotone SOCCP. For example, under the more general setting, every regularized problem has the unique solution, and the solution trajectory generated is bounded if the original SOCCP has a nonempty and bounded solution set. We also propose an inexact regularization algorithm by solving the sequence of regularized problems approximately with the merit function approach based on Fischer–Burmeister merit function, and establish the convergence result of the algorithm. Preliminary numerical results are also reported, which verify the favorable theoretical properties of the proposed method.

A merit function approach to the subgradient method with averaging

We consider a version of the subgradient method for convex nonsmooth optimization involving subgradient averaging. Using a merit function approach in the space of decisions and subgradient estimates, we prove convergence of the primal variables to an optimal solution and of the dual variables to an optimal subgradient. Application to dual convex optimization problems is discussed.

A Class of Interior Proximal-Like Algorithms for Convex Second-Order Cone Programming

We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on $(0,+\infty)$, and includes as a special case the entropy-like proximal algorithm [Eggermont, Linear Algebra Appl., 130 (1990), pp. 25–42], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the sequence generated is bounded, and every accumulation point is a solution of the considered problem. Preliminary numerical results are reported for two approximate entropy-like proximal algorithms, and numerical comparisons are also made with the merit function approach [Chen and Tseng, Math. Program., 104 (2005), pp. 293–327], which verify the effectiveness of the proposed method.

A descent method for a reformulation of the second-order cone complementarity problem

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over R n . In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the Fischer–Burmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327]. The comparison results indicate that, if a scaling strategy is imposed on the test problem, the descent method proposed is comparable with the merit function approach in the CPU time for solving test problems although the former may require more function evaluations.

On implementation of a self-dual embedding method for convex programming

In this paper, we implement Zhang's method [Zhang, S., 2004, Journal of Global Optimization, 29, 479–496], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of self-dual embedding and central path-following for solving the resulting conic optimization model. A crucial advantage of the approach is that no initial solution is required and the method is particularly suitable when the feasibility status of the problem is unknown. In our implementation, we use a merit function approach proposed by Andersen and Ye [Andersen, E.D. and Ye, Y., 1998, Computational Optimization and Applications, 10, 243–269] to determine the step-size along the search direction. We evaluate the efficiency of the proposed algorithm by observing its performance on some test problems, which include logarithmic functions, exponential functions and quadratic functions in the constraints. Furthermore, we consider in particular the geometric programming and L p -programming problems. Numerical results of our algorithm on these classes of optimization problems are reported. We conclude that the algorithm is stable, efficient and easy-to-use in general. As the method allows the user to freely select the initial solution, it is natural to take advantage of this and apply the so-called warm-start strategy, whenever the data of a new problem is not too much different from a previously solved problem. This strategy turns out to be effective, according to our numerical experience.

Cartesian P-property and Its Applications to the Semidefinite Linear Complementarity Problem

We introduce a Cartesian P-property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem (SDLCP). With this Cartesian P-property, we show that the SDLCP has GUS-property (i.e., globally unique solvability), and the solution map of the SDLCP is locally Lipschitzian with respect to input data. Our Cartesian P-property strengthens the corresponding P-properties of Gowda and Song [15] and allows us to extend several numerical approaches for monotone SDLCPs to solve more general SDLCPs, namely SDLCPs with the Cartesian P-property. In particular, we address important theoretical issues encountered in those numerical approaches, such as issues related to the stationary points in the merit function approach, and the existence of Newton directions and boundedness of iterates in the non-interior continuation method of Chen and Tseng [6].