Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems

Summary. A new numerical algorithm for solving semilinear elliptic problems is presented. A variational formulation is used and critical points of a C¹-functional subject to a constraint given by a level set of another C¹-functional (or an intersection of such level sets of finitely many functionals) are sought. First, constrained local minima are looked for, then constrained mountain pass points. The approach is based on the deformation lemma and the mountain pass theorem in a constrained setting. Several examples are given showing new numerical solutions in various applications.Mathematics Subject Classification (2000):35J20, 65N99The author would like to thank the referee for helpful comments in particular on Section 4.     

A shape and topology optimization technique for solving a class of linear complementarity problems in function space

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive set, the original problem is reformulated as a shape optimization problem. The topological sensitivity of the new objective functional is used to estimate the “topology” of the active set. Then, for local correction purposes near the interface, a level set based shape sensitivity technique is employed. A numerical algorithm is devised, and a report on numerical test runs ends the paper.     

Scalarizations for adaptively solving multi-objective optimization problems

In this paper several parameter dependent scalarization approaches for solving nonlinear multi-objective optimization problems are discussed. It is shown that they can be considered as special cases of a scalarization problem by Pascoletti and Serafini (or a modification of this problem). Based on these connections theoretical results as well as a new algorithm for adaptively controlling the choice of the parameters for generating almost equidistant approximations of the efficient set, lately developed for the Pascoletti-Serafini scalarization, can be applied to these problems. For instance for such well-known scalarizations as the ε-constraint or the normal boundary intersection problem algorithms for adaptively generating high quality approximations are derived.     

A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems

In this paper we propose a new algorithm for solving difficult large-scale global optimization problems. We draw our inspiration from the well-known DIRECT algorithm which, by exploiting the objective function behavior, produces a set of points that tries to cover the most interesting regions of the feasible set. Unfortunately, it is well-known that this strategy suffers when the dimension of the problem increases. As a first step we define a multi-start algorithm using DIRECT as a deterministic generator of starting points. Then, the new algorithm consists in repeatedly applying the previous multi-start algorithm on suitable modifications of the variable space that exploit the information gained during the optimization process. The efficiency of the new algorithm is pointed out by a consistent numerical experimentation involving both standard test problems and the optimization of Morse potential of molecular clusters.     

CHESS—Changing Horizon Efficient Set Search: A Simple Principle for Multiobjective Optimization

This paper presents a new concept for generating approximations to the non-dominated set in multiobjective optimization problems. The approximation set A is constructed by solving several single-objective minimization problems in which a particular function D(A, z) is minimized. A new algorithm to calculate D(A, z) is proposed.No general approach is available to solve the one-dimensional optimization problems, but metaheuristics based on local search procedures are used instead. Tests with multiobjective combinatorial problems whose non-dominated sets are known confirm that CHESS can be used to approximate the non-dominated set. Straightforward parallelization of the CHESS approach is illustrated with examples.The algorithm to calculate D(A, z) can be used in any other applications that need to determine Tchebycheff distances between a point and a dominant-free set.     

A posteriori error estimation for finite element discretization of parameter identification problems

Summary. In this paper we develop an a posteriori error estimator for parameter identification problems. The state equation is given by a partial differential equation involving a finite number of unknown parameters. The presented error estimator aims to control the error in the parameters due to discretization by finite elements. For this, we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence. Exploiting the special structure of the parameter identification problem, allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm. Several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section. For the problems considered here, both, the efficiency of the estimator and the quality of the generated meshes are satisfactory. Mathematics Subject Classifications (2000): 65K10, 65N30, 49K20This work has been supported by the German Research Foundation (DFG) through SFB 359 Reactive Flow, Diffusion and Transport and Graduiertenkolleg Modellierung und wissenschaftliches Rechnen in Mathematik und Naturwissenschaften     

A fast steady-state <Emphasis Type="Italic">ε</Emphasis>-dominance multi-objective evolutionary algorithm

Multi-objective evolutionary algorithms (MOEAs) have become an increasingly popular tool for design and optimization tasks in real-world applications. Most of the popular baseline algorithms are pivoted on the use of Pareto-ranking (that is empirically inefficient) to improve the convergence to the Pareto front of a multi-objective optimization problem. This paper proposes a new ε-dominance MOEA (EDMOEA) which adopts pair-comparison selection and steady-state replacement instead of the Pareto-ranking. The proposed algorithm is an elitist algorithm with a new preservation technique of population diversity based on the ε-dominance relation. It is demonstrated that superior results could be obtained by the EDMOEA compared with other algorithms: NSGA-II, SPEA2, IBEA, ε-MOEA, PESA and PESA-II on test problems. The EDMOEA is able to converge to the Pareto optimal set much faster especially on the ZDT test functions with a large number of decision variables.     

Resolution method for mixed integer bi-level linear problems based on decomposition technique

In this article, we propose a new algorithm for the resolution of mixed integer bi-level linear problem (MIBLP). The algorithm is based on the decomposition of the initial problem into the restricted master problem (RMP) and a series of problems named slave problems (SP). The proposed approach is based on Benders decomposition method where in each iteration a set of variables are fixed which are controlled by the upper level optimization problem. The RMP is a relaxation of the MIBLP and the SP represents a restriction of the MIBLP. The RMP interacts in each iteration with the current SP by the addition of cuts produced using Lagrangian information from the current SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is within a small difference ε. In the case of MIBLP Karush–Kuhn–Tucker (KKT) optimality conditions could not be used directly to the inner problem in order to transform the bi-level problem into a single level problem. The proposed decomposition technique, however, allows the use of KKT conditions and transforms the MIBLP into two single level problems. The algorithm, which is a new method for the resolution of MIBLP, is illustrated through a modified numerical example from the literature. Additional examples from the literature are presented to highlight the algorithm convergence properties.     

A characterization of convex calibrable sets in

The main purpose of this paper is to characterize the calibrability of bounded convex sets in by the mean curvature of its boundary, extending the known analogous result in dimension 2. As a by-product of our analysis we prove that any bounded convex set C of class C^1,1 has a convex calibrable set K in its interior, and and for any volume V [|K|,|C|] the solution of the perimeter minimizing problem with fixed volume V in the class of sets contained in C is a convex set. As a consequence we describe the evolution of convex sets in by the minimizing total variation flow.Mathematics Subject Classification (2000): 35J70, 49J40, 52A20, 35K65     

Combining search directions using gradient flows

 The efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations to the Newton direction. In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection. Received: November 2000 / Accepted: October 2002 Published online: February 14, 2003 Key Words. negative curvature – primal-dual methods – interior-point methods – nonconvex optimization – line searches Mathematics Subject Classification (1991): 49M37, 65K05, 90C30     

Hydrodynamic limit for ∇φ interface model on a wall

 We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational inequality. Received: 10 January 2002 / Revised version: 18 August 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85 Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational inequality     

Applying the Powell's Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition

This article proposes new conjugate gradient method for unconstrained optimization by applying the Powell symmetrical technique in a defined sense. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. Preliminary numerical results for a set of 86 unconstrained optimization test problems verify the performance of the algorithm and show that the Generalized Descent Symmetrical Hestenes-Stiefel algorithm is competitive with the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP⁺) algorithms.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

Optimality Conditions for the Difference of Convex Set-Valued Mappings

Using the concept of subdifferential of cone-convex set valued mappings recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we give necessary optimality conditions for nonconvex multiobjective optimization problems. An example illustrating the usefulness of our results is also given. Mathematics Subject classification: Primary 90C29, 90C26; Secondary 49K99.     

Lipschitz continuity of state functions in some optimal shaping

We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves.Received: 11 November 2003, Accepted: 10 May 2004, Published online: 8 February 2005Mathematics Subject Classification (2000): 49Q10, 35R35, 49K20, 35J20     

Accelerated conjugate gradient algorithm with finite difference Hessian/vector product approximation for unconstrained optimization

In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter β_k is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

Stability in disjunctive linear optimization I: continuity of the feasible set

The aim of this paper is to study the continuous dependence of the feasible set of a disjunctive semi-infinite linear optimization problem on all involved parameters (matrix and right-hand side). The feasible set of such an optimization problem is the union of (a. possible infinite number of) convex sets, which each is described by a finite or an infinite number of strict and non-strict linear inequalities. We derive necessary and sufficient conditions for the upper- and lower-semi-continuity, and the closedness of the feasible-set-mapping Z Especially, the compactness of the boundary of the feasible set and the closedness of Z are equivalent to the upper-semi-continuity of Zwhile the lower semi-continuity of Z is equivalent to a certain constraint qualification. This constraint qualification is a strengthened kind of Slater condition, rrom tuese investigations, we derive known results in parametric semi-infinite optimization and parametric integer programming.     

Partial regularity results for evolutional p-Laplacian systems with natural growth

 We study a regularity for evolutional p-Laplacian systems with natural growth on the gradient. It is shown that weak solutions of small image and their gradients are partial H?lder continuous and the size of the exceptional set for regularity is estimated in terms of Hausdorff measure. The main ingredient is to improve the Gehring inequality, which implies the higher integrability of the gradient and was first developed by Kinnunen and Lewis, so as to be well-worked in our perturbation estimate. We also use a refinement of the perturbation argument and make a device for H?lder estimates of the gradient. Received: 4 March 2002 Mathematics Subject Classification (2000): Primary 35D10, 35B65, 35K65