A dual differentiable exact penalty function

A new penalty function is associated with an inequality constrained nonlinear programming problem via its dual. This penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable. In addition, the penalty parameter remains finite. This approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a Euclidean space and the nonnegative orthant of another Euclidean space. Many efficient algorithms exist for solving this problem. For the case of quadratic programming, the penalty function problem can be solved effectively by successive overrelaxation (SOR) methods which can handle huge problems while preserving sparsity features. Sponsored by the United States Army under Contract No. DAAG 29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants No. MCS-790166 and ENG-7903881.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

A lower bound for the controlling parameters of the exact penalty functions

The purpose of this paper is to present new exact penalty functions and discuss their properties. A lower bound on the controlling parameters is given, for which above this value, the optimum of the exact penalty function coincides with the optimum of the nonlinear programming problem.This work was supported by the National Research Council of Canada under Grant A4414.     

Stable exponential-penalty algorithm with superlinear convergence

A renewed interest in penalty algorithms for solving mathematical programming problems has been motivated by some recent techniques which eliminate the ill-conditioning caused by the convergence to zero of the penalty parameter. These techniques are based on a good identification of the active set of constrainst at the optimum. In this sense, interior penalty methods to be more efficient than exterior ones, but their drawback lies in the need of an interior starting point. We propose in this paper an exponential penalty function which does not need interior starting points, but whose ultimate behavior is just like an interior penalty method. A superlinearly convergent algorithm based on the exponential penalty function is proposed.This research was partially supported by FONDECYT Grant 90-0945, DTI Grant E.3101-9012, and NSERC Grant OGP0005491.     

Second-order conditions for an exact penalty function

In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related.In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.This research is partially supported by the Natural Science and Engineering Research Council Grant No. A8639 and the U.S. Department of Energy.     

非光滑多目标最优化的一个精确罚函数

本文将文献[1]中的关于单目标规划的精确罚函数的结果推广到局部Lipschitz多目标规划问题上.     

Existence of exact penalty for optimization problems with mixed constraints in Banach spaces

In this paper we use the penalty approach in order to study constrained minimization problems in a Banach space with nonsmooth nonconvex mixed constraints. A penalty function is said to have the exact penalty property [J.-B. Hiriart-Urruty, C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993] if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish sufficient conditions for the exact penalty property.     

Linear programming via a quadratic penalty function

We use quadratic penalty functions along with some recent ideas from linearl ₁ estimation to arrive at a new characterization of primal optimal solutions in linear programs. The algorithmic implications of this analysis are studied, and a new, finite penalty algorithm for linear programming is designed. Preliminary computational results are presented.Research supported by grant No. 11-0505 from the Danish Natural Science Research Council SNF.     

Nonlinear programming via an exact penalty function: Global analysis

In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.This work is supported in part by NSERC Grant No. A8639 and the U.S. Dept. of Energy.     

Interval goal programming for S-shaped penalty function

The approach of Jones and Tamiz (1995) [Jones, D.F., Tamiz, M., 1995. Expanding the flexibility of goal programming via preference modeling techniques. Omega 23, 41–48] has been accepted as the most efficient approach in the field of interval goal programming (IGP). Although several modifications to the original approach have been proposed recently [Vitoriano, B., Romero, C., 1999. Extended interval goal programming. Journal of the Operational Research Society 50, 1280–1283; Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396], all of them cannot formulate IGP with an S-shaped penalty function. In order to improve the utility of IGP, we extend the model of Chang (2006) [Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396] to be able to model an S-shaped penalty function. The newly formulated model is more concise and compact than the method of Li and Yu (2000) and it can easily be applied to a decision problem with the S-shaped penalty function. Finally, an illustrative example (i.e. how to build an appropriate E-learning system) is included for demonstrating the usefulness of the proposed model.     

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In this paper, we put non-concave penalty on the local conditional likelihood. We obtain the oracle property and asymptotic normal distribution property of the parameters in Ising model. With a union band, we obtain the sign consistence for the estimator of parameter matrix, and the convergence speed under the matrix $L_1$ norm. The results of the simulation studies and a real data analysis show that the non-concave penalized estimator has larger sensitivity.     

Optimization problems for an energy functional with mass constraint revisited

In this paper we report on a variational problem under a constraint on the mass which is motivated by the torsional rigidity and torsional creep. Following a device by Alt, Caffarelli and Friedman we treat instead a problem without constraint but with a penalty term. We will complete some of the results of [C. Bandle, A. Wagner, Optimization problems for weighted Sobolev constants, Calc. Var. Partial Differential Equations 29 (2007) 481-507] where the existence of a Lipschitz continuous minimizer has been established. In particular we prove qualitative properties of the optimal shape.     

Exact barrier function methods for Lipschitz programs

The purpose of this paper is twofold. First we consider a class of nondifferentiable penalty functions for constrained Lipschitz programs and then we show how these penalty functions can be employed to solve a constrained Lipschitz program. The penalty functions considered incorporate a barrier term which makes their value go to infinity on the boundary of a perturbation of the feasible set. Exploiting this fact it is possible to prove, under mild compactness and regularity assumptions, a complete correspondence between the unconstrained minimization of the penalty functions and the solution of the constrained program, thus showing that the penalty functions are exact according to the definition introduced in [17]. Motivated by these results, we propose some algorithm models and study their convergence properties. We show that, even when the assumptions used to establish the exactness of the penalty functions are not satisfied, every limit point of the sequence produced by a basic algorithm model is an extended stationary point according to the definition given in [8]. Then, based on this analysis and on the one previously carried out on the penalty functions, we study the consequence on the convergence properties of increasingly demanding assumptions. In particular we show that under the same assumptions used to establish the exactness properties of the penalty functions, it is possible to guarantee that a limit point at least exists, and that any such limit point is a KKT point for the constrained problem.This research has been partially supported by the National Research Program on Metodi di Ottimizzazione per le Decisioni, Ministero dell' Università e della Ricerca Scientifica e Tecnologica.     

Existence of Classical Solution of a Quasilinear Stefan-Signorini Problem

In this paper, we consider an one-phase quasilinear Stefan-Signorini problem. In [1] X. Liu has proved the existence and the uniqueness of this problem by the traditional penalty method (see [3] and [4]). Here we use the Mixed Scheme in [2] which is more suitable to the Stefan-Signorini problem and obtain more natural existence results.     

四阶超线性奇异微分方程正解存在的充分必要条件

利用范数形式的锥拉伸与压缩不动点定理,对一类四阶奇异超线性微分方程边值问题做了研究,得到C~2[0,1]正解与C~3[0,1]正解存在的充分必要条件,也得到C~2[0,1]正解的不可比较性等解的性质.     

Homogenization of some degenerate pseudoparabolic variational inequalities

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.     

A power penalty approach to a mixed quasilinear elliptic complementarity problem

In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust.

     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

Generalized complementarity problem

A general complementarity problem with respect to a convex cone and its polar in a locally convex, vector-topological space is defined. It is observed that, in this general setting, the problem is equivalent to a variational inequality over a convex cone. An existence theorem is established for this general case, from which several of the known results for the finite-dimensional cases follow under weaker assumptions than have been required previously. In particular, it is shown that, if the given map under consideration is strongly copositive with respect to the underlying convex cone, then the complementarity problem has a solution.This research was partially supported by the Office of Naval Research, Contract No. N-00014-67-A0112-0011, by the Atomic Energy Commission, Contract No. AT[04-3]-326-PA-18, and by the National Science Foundation, Grant No. GP-9329.