scholarly journals A new approach to optimality in a class of nonconvex smooth optimization problems

2018 ◽  
Vol 34 (1) ◽  
pp. 01-07
Author(s):  
TADEUSZ ANTCZAK ◽  
Keyword(s):  
Approximation Method ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Extremum Problem ◽  
Optimal Solutions ◽  
New Approach ◽  
Smooth Optimization

In this paper, a new approximation method for a characterization of optimal solutions in a class of nonconvex differentiable optimization problems is introduced. In this method, an auxiliary optimization problem is constructed for the considered nonconvex extremum problem. The equivalence between optimal solutions in the considered differentiable extremum problem and its approximated optimization problem is established under (Φ, ρ)-invexity hypotheses.

Adaptive Grouping Brain Storm Optimization for Multimodal Optimization Problems

2021 ◽  
Vol 12 (4) ◽  
pp. 81-100
Author(s):  
Yao Peng ◽  
Zepeng Shen ◽  
Shiqi Wang
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Algorithm ◽  
Peak Detection ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Grouping Strategy ◽  
Different Dimensions ◽  
Global Optimal ◽  
Localization Ability

Multimodal optimization problem exists in multiple global and many local optimal solutions. The difficulty of solving these problems is finding as many local optimal peaks as possible on the premise of ensuring global optimal precision. This article presents adaptive grouping brainstorm optimization (AGBSO) for solving these problems. In this article, adaptive grouping strategy is proposed for achieving adaptive grouping without providing any prior knowledge by users. For enhancing the diversity and accuracy of the optimal algorithm, elite reservation strategy is proposed to put central particles into an elite pool, and peak detection strategy is proposed to delete particles far from optimal peaks in the elite pool. Finally, this article uses testing functions with different dimensions to compare the convergence, accuracy, and diversity of AGBSO with BSO. Experiments verify that AGBSO has great localization ability for local optimal solutions while ensuring the accuracy of the global optimal solutions.

Space Contraction and Partition Algorithm for Combinatorial Optimization

2011 ◽  
Vol 421 ◽  
pp. 559-563
Author(s):  
Yong Chao Gao ◽  
Li Mei Liu ◽  
Heng Qian ◽  
Ding Wang
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Solution Space ◽  
Search Space ◽  
Small Scale ◽  
Optimal Solutions ◽  
Solution Quality ◽  
Intelligent Algorithms ◽  
Combinatorial Optimization Problems

The scale and complexity of search space are important factors deciding the solving difficulty of an optimization problem. The information of solution space may lead searching to optimal solutions. Based on this, an algorithm for combinatorial optimization is proposed. This algorithm makes use of the good solutions found by intelligent algorithms, contracts the search space and partitions it into one or several optimal regions by backbones of combinatorial optimization solutions. And optimization of small-scale problems is carried out in optimal regions. Statistical analysis is not necessary before or through the solving process in this algorithm, and solution information is used to estimate the landscape of search space, which enhances the speed of solving and solution quality. The algorithm breaks a new path for solving combinatorial optimization problems, and the results of experiments also testify its efficiency.

scholarly journals ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhi-Ang Zhou
Keyword(s):  
Saddle Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Saddle Points ◽  
Linear Spaces ◽  
Equivalent Characterization ◽  
Generalized Cone Subconvexlikeness ◽  
Real Linear ◽  
The Relationship

We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization ofϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between theϵ-Henig saddle point of the Lagrangian set-valued map and theϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.

scholarly journals Efficiently Sparse Listing of Classes of Optimal Cophylogeny Reconciliations

2021 ◽  
Author(s):  
Yishu Wang ◽  
Arnaud Mary ◽  
Marie-France Sagot ◽  
Blerina Sinaimeri
Keyword(s):  
Optimization Problem ◽  
Practical Importance ◽  
Biological Information ◽  
Optimal Solutions ◽  
Powerful Method ◽  
New Approach ◽  
Significant Research ◽  
Whole Space ◽  
Polynomial Delay ◽  
Host Parasite

Abstract Background: Cophylogeny reconciliation is a powerful method for analyzing host-parasite (or host-symbiont) co-evolution. It models co-evolution as an optimization problem where the set of all optimal solutions may represent different biological scenarios which thus need to be analyzed separately. Despite the significant research done in the area, few approaches have addressed the problem of helping the biologist deal with the often huge space of optimal solutions.Results: In this paper, we propose a new approach to tackle this problem. We introduce three different criteria under which two solutions may be considered biologically equivalent, and then we propose polynomial-delay algorithms that enumerate only one representative per equivalence class (without listing all the solutions).Conclusions: Our results are of both theoretical and practical importance. Indeed, as shown by the experiments, we are able to significantly reduce the space of optimal solutions while still maintaining important biological information about the whole space.

scholarly journals Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

2021 ◽  
Vol 8 ◽  
Author(s):  
Tilo Strutz
Keyword(s):  
Computational Complexity ◽  
Optimization Problems ◽  
State Of The Art ◽  
Travelling Salesman Problem ◽  
Limited Time ◽  
Two Dimensional ◽  
Optimal Solutions ◽  
Major Difficulty ◽  
Travelling Salesman ◽  
New Approach

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist for many problems up to several ten thousand points. The major difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-art methods when applied to TSP instances with non-uniformly distributed coordinates.

A Boolean Local Improvement Method for General Discrete Optimization Problems

1993 ◽  
Vol 115 (4) ◽  
pp. 776-783 ◽  
Author(s):  
Peng Lu ◽  
J. N. Siddall
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Continuous Variable ◽  
Local Region ◽  
Boolean Logic ◽  
Discrete Variable ◽  
New Approach ◽  
Local Improvement ◽  
Improvement Method ◽  
Discrete Optimization Problems

The paper proposes a new approach to solving the discrete design optimization problem by combining Boolean logic methods with conventional nonlinear optimization. A continuous variable optimum is first found, and then the Boolean method searches the local region for the discrete variable optimum.

Optimality Conditions for Pareto-Optimal Solutions

2016 ◽  
Vol 685 ◽  
pp. 142-147
Author(s):  
Vladimir Gorbunov ◽  
Elena Sinyukova
Keyword(s):  
Multicriteria Optimization ◽  
Optimization Problem ◽  
Necessary Conditions ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Sufficient Conditions ◽  
Optimal Solutions ◽  
Main Criterion ◽  
Linear Optimization Problems ◽  
Necessary And Sufficient

In this paper the authors describe necessary conditions of optimality for continuous multicriteria optimization problems. It is proved that the existence of effective solutions requires that the gradients of individual criteria were linearly dependent. The set of solutions is given by system of equations. It is shown that for finding necessary and sufficient conditions for multicriteria optimization problems, it is necessary to switch to the single-criterion optimization problem with the objective function, which is the convolution of individual criteria. These results are consistent with non-linear optimization problems with equality constraints. An example can be the study of optimal solutions obtained by the method of the main criterion for Pareto optimality.

scholarly journals On G-invexity-type nonlinear programming problems

2015 ◽  
Vol 5 (1) ◽  
pp. 13-20
Author(s):  
Tadeusz Antczak ◽  
Manuel Arana Jiménez
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Extremum Problem ◽  
Sufficient Optimality Conditions ◽  
New Class ◽  
Necessary And Sufficient ◽  
Wolfe Dual

In this paper, we introduce the concepts of KT-G-invexity and WD$-G-invexity for the considered differentiable optimization problem with inequality constraints. Using KT-G-invexity notion, we prove new necessary and sufficient optimality conditions for a new class of such nonconvex differentiable optimization problems. Further, the so-called G-Wolfe dual problem is defined for the considered extremum problem with inequality constraints. Under WD-G-invexity assumption, the necessary and sufficient conditions for weak duality between the primal optimization problem and its G-Wolfe dual problem are also established.

scholarly journals Optimization problems, first order approximated optimization problems and their connections

2012 ◽  
Vol 28 (1) ◽  
pp. 133-141
Author(s):  
EMILIA-LOREDANA POP ◽  
◽  
DOREL I. DUCA ◽  
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Saddle Points ◽  
Optimal Solutions ◽  
First Order

In this paper, we attach to the optimization problem ... where X is a subset of Rn, f : X → R, g = (g1, ..., gm) : X → Rm and h = (h1, ..., hq) : X → Rq are functions, the (0, 1) − η− approximated optimization problem (AP). We will study the connections between the optimal solutions for Problem (AP), the saddle points for Problem (AP), optimal solutions for Problem (P) and saddle points for Problem (P).

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