Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.

This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.

We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions.

Abstract. We study the asymptotic behavior of the statistical estimators that maximize a not necessarily differentiable criterion function, possibly subject to side constraints (equalities and inequalities). The consistency results generalize those of Wald and Huber. Conditions are also given under which one is still able to obtain asymptotic normality. The analysis brings to the fore the relationship between the problem of finding statistical estimators and that of finding the optimal solutions of stochastic optimization problems with partial information. The last section is devoted to the properties of the saddle points of the associated Lagrangians.

Abstract. In this paper we generalize the iterated refinement method, introduced by the authors in [8], to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation. 1

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. Various search directions used in interior-point-algorithms for the SDP (semidefinite program) and the monotone SDLCP (semidefinite linear complementarity problem) are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace. This observation provides a unified geometric view over the existence of those search directions. Key words Interior-Point Algorithm, Semidefinite Program, Semidefinite Linear Complementarity Problem, Monotonicity y Department of Mathematics, Kanagawa University, Rokkakubashi 3-27-1, Kanagawa-ku, Yokohama 221, Japan. z Department of Mathematics and Physics, The National Defense Academy, Hashirimizu 1-10-20, Yokosuka, 239, Japan. ] Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan. Research Report B-310, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan 1. Introduction. ...

Most minimax theorems in critical point theory require one to solve a two-level global optimization problem and therefore are not for algorithm implementation. The objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way. In this paper, inspired by the numerical works of Choi-McKenna and Ding-Costa-Chen, and the idea to define a solution submanifold, some local minimax theorems are established, which require to solve only a two-level local optimization problem. Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed. The local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some non-convex, non star-shaped and multi-connected domains. Numerical solutions are illustrated by their graphics for visualization. In a subsequent paper [20], we establish some convergence results for the algorithm.

The concept of a monotone operator — which covers both linear positive semi-definite operators and subdifferentials of convex functions — is fundamental in various branches of mathematics. Over the last few decades, several stronger notions of monotonicity have been introduced: Gossez’s maximal monotonicity of dense type, Fitzpatrick and Phelps’s local maximal monotonicity, and Simons’s monotonicity of type (NI). While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for subdifferentials of convex functions, their precise relationship is largely unknown. Here, it is shown — within the beautiful framework of Convex Analysis — that for continuous linear monotone operators, all these notions coincide and are equivalent to the monotonicity of the conjugate operator. This condition is further

this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain--pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical e#ects of the domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial di#erential equations include the Lane--Emden equation, Chandrasekhar's equation, Henon's equation, a singularly perturbed equation, and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented