On local Pareto optima

Wan, Yieh-Hei

doi:10.1016/0304-4068(75)90012-9

Cited by 48 publications

(12 citation statements)

References 2 publications

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“…Differential approach to vector optimization dates back to Frisch (1966). As in the classical calculus, we deal here with first-order necessary and second-order sufficient conditions; for Pareto optima those were developed by Smale ( , 1975b and Wan (1975Wan ( , 1978. Their first-order, necessary condition for a Pareto optimum turns out to be actually the same as the one required for a weak optimum; see 5.1 for more details.…”

Section: Examplesmentioning

confidence: 99%

On depth and deep points: a calculus

Mizera¹

2002

Ann. Statist.

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Abstract. For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of linear functions in the mathematical analysis. As an application, a lower bound for maximal regression depth is proved in the general multidimensional case-as conjectured by Rousseeuw and Hubert, and others. This lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator.

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Section: Examplesmentioning

confidence: 99%

On depth and deep points: a calculus

Mizera¹

2002

Ann. Statist.

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“…Theorem 3.4 is in Smale 0974-76, V) and Wan (1975). See also Simon (forthcoming) for further information on this.…”

Section: Pareto Optimalitymentioning

confidence: 94%

Chapter 8 Global analysis and economics

Smale

1981

Handbook of Mathematical Economics Volume 1

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“…Global analysis and multiobjective nonlinear programming. With this in mind, we have devised a novel numerical strategy for approximating Pareto sets, theoretically based on the global analysis 4 framework established by Smale and others in the early 1970s [13], [14], [16], [17], [57], [58], [65], [66], [67] and in more recent work [35], [36]. Motivated by his discussions with Debreu, 5 Smale investigated the problem of optimizing several functions within the dynamical systems arena.…”

Section: Multiobjective Optimization and Pareto Optimality Multiobjementioning

confidence: 99%

Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis

Lovison¹

2011

SIAM J. Optim.

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Abstract. We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale [Global analysis and economics. I. Pareto optimum and a generalization of Morse theory, in Dynamical Systems, Academic Press, New York, 1973, pp. 531-544]. The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set, and stable Pareto critical set, and it can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples.

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On local Pareto optima

Cited by 48 publications

References 2 publications

On depth and deep points: a calculus

On depth and deep points: a calculus

Chapter 8 Global analysis and economics

Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis

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