Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

Data envelopment analysis (DEA) - Thirty years on

Pitfalls and protocols in DEA

This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

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An overview of bilevel optimization

A survey of data envelopment analysis in energy and environmental studies

Undesirable outputs in efficiency valuations

We study mathematical programs with complementarity constraints. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Further, we present Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima to mathematical programs with complementarity constraints.

Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

Data envelopment analysis (DEA) is a methodology that allows, in one way or other, the assignment of efficiency scores to members of a group of decision-making units. We call an efficiency measure "continuous" if small perturbations of the input-output data cause only small changes in the score. Continuity is a desirable property of an efficiency measure, in particular in the presence of measurement tolerances. Continuity is also desirable from a numerical point of view because the scores are computed by linear programming software.Focusing on convex production possibility sets, we give examples where radial DEA measures fail to be continuous, i.e., they "jump" under small data perturbations. We present necessary and sufficient conditions for continuity in terms of the data and show that these conditions are satisfied for "almost all" data. We also discuss continuity of nonradial measures and identify possible problems of "multistage approaches" to compute mix efficiencies.

Continuity of DEA Efficiency Measures

Continuity is a desirable property of an efficiency measure. It ensures that small data errors cause only small errors in the efficiency measure. In this paper continuity properties of the BCC efficiency measure are studied. Examples are given where this measure “jumps” under arbitrary small data perturbations. However, it is shown that under weak assumptions it depends continuously on the input output data. Implications to the stability of efficiency classifications are discussed, and the results are illustrated by an empirical example.

Holger Scheel

Papers

Undesirable outputs in efficiency valuations

Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

Continuity of DEA Efficiency Measures

Continuity of the BCC Efficiency Measure