Inexact subgradient methods for quasi-convex optimization problems

Hu, Yaohua; Yang, Xiaoqi; Sim, Chee-Khian

doi:10.1016/j.ejor.2014.05.017

Cited by 35 publications

(71 citation statements)

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“…Useful algorithms for solving Problem 2.1 were proposed in [11,13,22,23]. However, they assume that the metric projection onto the set X can be computed explicitly.…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

“…We call a functional of which any slice is convex a quasiconvex functional, and the class of this functional is a generalization of convex functionals. Quasiconvex functionals inherit some nice properties of convex functionals [13]. However, they do not have all the important properties of convex functionals, such as convexity of the sum of convex functionals, or give a guarantee of the coincidence of local optimality and global optimality.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, these problems can be dealt with as constrained quasiconvex optimizations. Here, we will examine the numerical behaviors of the existing and proposed algorithms when they are applied to the Cobb-Douglas production efficiency problem [3,Problem (3.13)], [11,Problem (6.1)], [35, Section 1.7], which is an instance of a fractional programming and constrained quasiconvex optimization problem. Furthermore, the demand for techniques to solve optimization problems is nowadays not only limited to convex objectives.…”

Section: Introductionmentioning

confidence: 99%

“…Kiwiel [22] proposed a subgradient method that uses a normalized normal vector to the slice as a subgradient (we will call it a subgradient throughout this paper) for solving the constrained quasiconvex optimization problem. Hu et al [11] analyzed its convergence properties when inexact subgradients are used and/or when it includes computational errors. Furthermore, a number of subgradient-method variants exist for solving quasiconvex optimization problems, such as the conditional subgradient methods [13] and the stochastic subgradient method [12].…”

Section: Introductionmentioning

confidence: 99%

“…Hu et al [11] analyzed its convergence properties when inexact subgradients are used and/or when it includes computational errors. Furthermore, a number of subgradient-method variants exist for solving quasiconvex optimization problems, such as the conditional subgradient methods [13] and the stochastic subgradient method [12].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Fixed point quasiconvex subgradient method

Hishinuma

Iiduka

2020

European Journal of Operational Research

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Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional objective functions, is an important instance. Subgradient methods and their variants are useful ways for solving these problems efficiently. Many complicated constraint sets onto which it is hard to compute the metric projections in a realistic amount of time appear in these applications. This implies that the existing methods cannot be applied to quasiconvex optimization over a complicated set. Meanwhile, thanks to fixed point theory, we can construct a computable nonexpansive mapping whose fixed point set coincides with a complicated constraint set. This paper proposes an algorithm that uses a computable nonexpansive mapping for solving a constrained quasiconvex optimization problem. We provide convergence analyses for constant diminishing step-size rules. Numerical comparisons between the proposed algorithm and an existing algorithm show that the proposed algorithm runs stably and quickly even when the running time of the existing algorithm exceeds the time limit.

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“…Useful algorithms for solving Problem 2.1 were proposed in [11,13,22,23]. However, they assume that the metric projection onto the set X can be computed explicitly.…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%