Local–Global Minimum Property in Unconstrained Minimization Problems

Burai, Pál

doi:10.1007/s10957-013-0432-3

Cited by 6 publications

(3 citation statements)

References 16 publications

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“…Their use is based on the local-global minimum property (see e.g. [10], [9] and the references therein). More precisely, all local minimizers of the previously listed functions are global minimizers as well.…”

Section: Applications and Special Casesmentioning

confidence: 99%

Hermite--Hadamard type inequality for certain Schur convex functions

Burai¹,

Makó²,

Szokol³

2019

Preprint

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Section: Applications and Special Casesmentioning

confidence: 99%

Hermite--Hadamard type inequality for certain Schur convex functions

Burai¹,

Makó²,

Szokol³

2019

Preprint

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“…As a byproduct, our function shows that the generalization of quasiconvexity to non-convex domains described in [6, Chapter 9] is a proper subset of global functions. This generalization was proposed in [41] and further investigated in [7,33,34,15,16,17]. It consists in replacing the segment used to define convexity and quasiconvexity by a continuous path.…”

Section: Notion Of Global Functionmentioning

confidence: 99%

A theory on the absence of spurious solutions for nonconvex and nonsmooth optimization

Josz¹,

Ouyang²,

Zhang³

et al. 2018

Preprint

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We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term global functions. They satisfy various powerful properties for analyzing nonconvex and nonsmooth optimization problems. For instance, they satisfy a theorem akin to the fundamental uniform limit theorem in the analysis regarding continuous functions. Global functions are also endowed with useful properties regarding the composition of functions and change of variables. Using these new results, we show that a class of nonconvex and nonsmooth optimization problems arising in tensor decomposition applications are global functions. This is the first result concerning nonconvex methods for nonsmooth objective functions. Our result provides a theoretical guarantee for the widely-used 1 norm to avoid outliers in nonconvex optimization.

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“…In general, for nonlinear problems, extrema can be located using first-order conditions, where the first derivative of the objective function is zero or undefined (critical points of a differentiable function). In a non-convex case, second-order conditions can then be used to determine whether the points in the solution space are locally optimal [6]; a matrix of second derivatives (Hessian Matrix) can be used for this purpose [7]. Although not applicable to every type of optimisation problem, KarshKuhn-Tucker conditions can be used to determine the necessary optimality conditions of problems with equality or inequality constraints.…”

Section: Introductionmentioning

confidence: 99%

Optimisation in Forest Management

et al. 2016

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The field of forestry has employed various computer-assisted optimisation approaches since the early 1960s to address the efficient allocation of resources towards various forest management objectives. These approaches continue to evolve, and in the last 5 years, the research has expanded to demonstrate how complex, non-linear relationships can be recognised and incorporated into planning processes at the tree, stand, forest and landscape levels. In addition to an overview of the use of optimisation in forestry, we provide an examination of work published in the last 5 years from 30 international journals, worldwide, which consistently publish forestry and natural resource management research papers. Through this review, we found that landscape-level optimisation is a relatively new and expanding area of research, most often performed by one large public landowner in regions where the resulting plan of action has an effect on all landowners and resources. We also note that at the forest level, exact methods for optimising systems mainly continue to be used, and at the stand level, optimisation seems to now involve exploration of a variety of analytical methods. A large portion of the recent research in the optimisation of forest management have involved European forests, which is a function of large public ownership of land and the tradition and requirements for management planning, and roughly half of the effort has arisen from researchers located in Nordic countries (Denmark, Finland, Norway and Sweden).

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Local–Global Minimum Property in Unconstrained Minimization Problems

Abstract: The main goal of this paper is to prove some new results and extend some earlier ones about functions, which possess the so called local-global minimum property. In the last section, we show an application of these in the theory of calculus of variations.

Cited by 6 publications

References 16 publications

Hermite--Hadamard type inequality for certain Schur convex functions

Hermite--Hadamard type inequality for certain Schur convex functions

A theory on the absence of spurious solutions for nonconvex and nonsmooth optimization

Optimisation in Forest Management

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