Log-Sigmoid nonlinear Lagrange method for nonlinear optimization problems over second-order cones

This paper analyzes the rate of local convergence of the Log-Sigmoid nonlinear Lagrange method for nonconvex nonlinear second-order cone programming. Under the componentwise strict complementarity condition, the constraint nondegeneracy condition and the second-order sufficient condition, we show that the sequence of iteration points generated by the proposed method locally converges to a local solution when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Finally, we report numerical results to show the efficiency of the method.     

On constraint qualifications in nonlinear programming

In this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one. When the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity, such conditions collapse into the well known constraint qualifications.     

BFGS trust-region method for symmetric nonlinear equations

In this paper, we propose a BFGS trust-region method for solving symmetric nonlinear equations. The global convergence and the superlinear convergence of the presented method will be established under favorable conditions. Numerical results show that the new algorithm is effective.     

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a P₀-matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in ℜⁿ.     

Convergence analysis of nonsmooth equations for the general nonlinear complementarity problem

This paper deals with a general nonlinear complementarity problem, where the underlying functions are assumed to be continuous. Based on a nonlinear complementarity function, it is transformed into a system of nonsmooth equations. Then, two kinds of approximate Newton methods for the nonsmooth equations are developed and their convergence are proved. Finally, numerical tests are also listed.     

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.     

A filled function method for global optimization

A novel filled function with one parameter is suggested in this paper for finding a global minimizer for a general class of nonlinear programming problems with a closed bounded box. A new algorithm is presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with (F,ρ)-convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F-approximated vector optimization problem, the suitable (F,ρ)-convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.     

Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds

Local convergence analysis of the proximal point method for a special class of nonconvex functions on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

A filled function method applied to nonsmooth constrained global optimization

The filled function method is an effective approach to find a global minimizer. In this paper, based on a new definition of the filled function for nonsmooth constrained programming problems, a one-parameter filled function is constructed to improve the efficiency of numerical computation. Then a corresponding algorithm is presented. It is a global optimization method which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method.     

A generalized Newton method for absolute value equations associated with second order cones

In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.     

Bilevel multiplicative problems: A penalty approach to optimality and a cutting plane based algorithm

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterised by the existence of two optimisation problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimisation problem. In this paper we focus on the class of bilevel problems in which the upper level objective function is linear multiplicative, the lower level one is linear and the common constraint region is a bounded polyhedron. After replacing the lower level problem by its Karush–Kuhn–Tucker conditions, the existence of an extreme point which solves the problem is proved by using a penalty function approach. Besides, an algorithm based on the successive introduction of valid cutting planes is developed obtaining a global optimal solution. Finally, we generalise the problem by including upper level constraints which involve both level variables.     

Linear bilevel programs with multiple objectives at the upper level

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making. When all objective functions are linear and constraints at both levels define polyhedra, it is proved that the set of efficient solutions is non-empty. Taking into account the properties of the feasible region of the bilevel problem, some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques. All the methods result in solving linear bilevel problems with a single objective function at each level.     

Linear bilevel programming with interval coefficients

In this paper, we address linear bilevel programs when the coefficients of both objective functions are interval numbers. The focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values. We prove by examples that, in general, there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions. Taking into account the properties of linear bilevel problems, we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints. Moreover, we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range.     

On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ℝ ⁿ to ℝ ^k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems. This work was supported by the Deutsche Forschungsgemeinschaft.     

Parameter estimation in systems of nonlinear equations

Summary. A numerical method is presented to determine parameters in a system of non\-linear equations in the following sense: Proceeding from given experimental data, e.g., observation times and measurements, the minimum least-squares distance of the measured data from a fitting criterion depending on the solution of a system of nonlinear equations is to be computed. Specifically coupled mass equilibrium models are described in more detail that are used in pharmaceutical applications for receptor-ligand binding studies. They are used for instance for the radioimmunological determination of Fenoterol or related substances. Numerical results based on laboratory data are included to test the robustness of the algorithm implemented. Received May 24, 1993/Revised version received February 13, 1994     

An augmented penalty function method with penalty parameter updates for nonconvex optimization

Given an augmented Lagrangian scheme for a general optimization problem, we use an epsilon subgradient step for improving the dual function. This can be seen as an update for an augmented penalty method, which is more stable because it does not force the penalty parameter to tend to infinity. We establish for this update primal-dual convergence for our augmented penalty method. As illustration, we apply our method to the test-bed kissing number problem.