Extended reverse-convex programming: an approximate enumeration approach to global optimization

Bunin, Gene A.

doi:10.1007/s10898-015-0352-x

Cited by 5 publications

(6 citation statements)

References 25 publications

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“…In Table 8, the solution x x x k of the from 4 to 6 columns is composed of x x x repeated p times. In Table 8, the values of the last six columns are obtained by Algorithm 1 (See Table 5,6,7). Numerical results show that for large-scale unconstrained optimization problems, a better solution can be obtained directly by Algorithm 1 by solving small-scale subproblems of it when the structure of all the subproblems are similar, i.e.…”

Section: Decomposable Algorithm Of (Cnp)mentioning

confidence: 99%

“…A new nonconvex function is defined in this paper, which is called the (weak or strong uniform) CN function in Definition 2.2, where the CN function is a nonconvex nonsmooth function form that can be transformed into a convex smooth function with convex equality constraints. The CN function somewhat relates to upper -UC k function [5,12,19,32,35,39] and factorable nonconvex function [6,17,24,25,29,30,41,36].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the factorable nonconvex functions in [25,29,30,41] may be special CN functions (see Definition 2.2). In recent years, research on nonconvex factorable programming further shows its effectiveness in solving the global optimization, as shown in [6,17,24,36].…”

Section: Introductionmentioning

confidence: 99%

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Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization

Jiang¹,

Shen²,

Meng³

et al. 2022

Preprint

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This paper defines a convertible nonconvex function(CN function for short) and a weak (strong) uniform (decomposable, exact) CN function, proves the optimization conditions for their global solutions and proposes algorithms for solving the unconstrained optimization problems with the decomposable CN function. First, to illustrate the fact that some nonconvex functions, nonsmooth or discontinuous, are actually weak uniform CN functions, examples are given. The operational properties of the CN functions are proved, including addition, subtraction, multiplication, division and compound operations. Second, optimization conditions of the global optimal solution to unconstrained optimization with a weak uniform CN function are proved. Based on the unconstrained optimization problem with the decomposable CN function, a decomposable algorithm is proposed by its augmented Lagrangian penalty function and its convergence is proved. Numerical results show that an approximate global optimal solution to unconstrained optimization with a CN function may be obtained by the decomposable algorithms. The decomposable algorithm can effectively reduce the scale in solving the unconstrained optimization problem with the decomposable CN function. This paper provides a new idea for solving unconstrained nonconvex optimization problems.

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Section: Decomposable Algorithm Of (Cnp)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization

Jiang¹,

Shen²,

Meng³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In this paper, we define a new nonconvex function (Definition 2.2 below), called the (weak or strong uniform) CN function, where the CN function is a nonconvex nonsmooth function form that can be transformed into a convex smooth function with convex equality constraints. The CN function somewhat relates to the upper -UC k function [5,11,17,28,31,34] and the factorable nonconvex function [6,15,21,22,25,26,36,32].…”

Section: Introductionmentioning

confidence: 99%

Optimization conditions and decomposable algorithms for convertible nonconvex optimization

2023

J. Nonlinear Var. Anal.

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This paper defines a convertible nonconvex function (CN function for short) and a weak (strong) uniform (decomposable, exact) CN function, proves the optimization conditions for their global solutions, and proposes algorithms for solving the unconstrained optimization problems with decomposable CN functions. First, to illustrate the fact that some nonconvex functions, nonsmooth or discontinuous, are actually weak uniform CN functions, examples are given. The operational properties of CN functions are proved, including addition, subtraction, multiplication, division, and compound operations. Second, optimization conditions of the global optimal solution to the unconstrained optimization with weak uniform CN function are proved. Based on the unconstrained optimization problem with decomposable CN functions, a decomposable algorithm is proposed by its augmented Lagrangian penalty function and its convergence is proved. Numerical results demonstrate that an approximate global optimal solution to unconstrained optimization with CN function may be obtained by the decomposable algorithm. The decomposable algorithm can effectively reduce the scale in solving the unconstrained optimization problem with decomposable CN function. This paper provides a new idea for solving unconstrained nonconvex optimization problems.

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“…A new nonconvex function is defined in this paper, which is called the SCN function in Definition 2.1, where the SCN function is a nonconvex nonsmooth function form that can be transformed into a convex smooth function with convex equality constraints. The SCN function somewhat relates to upper -UC k function [5,12,19,32,35,39] and factorable nonconvex function [6,17,24,25,29,30,41,36].…”

Section: Introductionmentioning

confidence: 99%

Exact Penalty Algorithm of Strong Convertible Nonconvex Optimization

Jiang¹,

Shen²,

Meng³

et al. 2022

Preprint

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This paper defines a strong convertible nonconvex(SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth(nondifferentiable) function. First, many examples of SCN function are given, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of the SCN functions are proved, including addition, multiplication, compound operations and so on. Third, the SCN forms of some special functions common in machine learning and engineering applications are presented respectively where these SCN function optimization problems can be transformed into minmax problems with a convex and concave objective function. Fourth,a minmax optimization problem of SCN function and its penalty function are defined. The optimization condition,exactness and stability of the minmax optimization problem are proved. Finally, an algorithm of penalty function to solve the minmax optimization problem and its convergence are given. This paper provides an efficient technique for solving unconstrained nonconvex or nonsmooth(nondifferentiable) optimization problems to avoid using subdifferentiation.

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Extended reverse-convex programming: an approximate enumeration approach to global optimization

Cited by 5 publications

References 25 publications

Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization

Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization

Optimization conditions and decomposable algorithms for convertible nonconvex optimization

Exact Penalty Algorithm of Strong Convertible Nonconvex Optimization

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