Characterizations of the solution set for a class of nonsmooth optimization problems

2012 ◽  
Vol 7 (4) ◽  
pp. 685-694 ◽  
Author(s):  
Ke Quan Zhao ◽  
Xin Min Yang
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Solution Set

scholarly journals Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Qinghai He ◽  
Weili Kong
Keyword(s):  
Banach Spaces ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Pareto Optimal ◽  
Pareto Solution ◽  
Value Set ◽  
Set Valued Mapping ◽  
Optimal Value

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

scholarly journals Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization

2019 ◽  
Vol 33 ◽  
pp. 2304-2313 ◽  
Author(s):  
Ken Kobayashi ◽  
Naoki Hamada ◽  
Akiyoshi Sannai ◽  
Akinori Tanaka ◽  
Kenichi Bannai ◽  
...  
Keyword(s):  
Sample Size ◽  
Optimization Problems ◽  
Solution Set ◽  
Small Samples ◽  
High Dimensional ◽  
Large Sample Size ◽  
Multi Objective Optimization ◽  
Dimensional Solution ◽  
Multi Objective ◽  
Simplex Model

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis.

Definition of the feasible solution set in multicriteria optimization problems with continuous, discrete, and mixed design variables

2009 ◽  
Vol 71 (12) ◽  
pp. e109-e117 ◽  
Author(s):  
Roman Statnikov ◽  
Alex Bordetsky ◽  
Josef Matusov ◽  
Il’ya Sobol’ ◽  
Alexander Statnikov
Keyword(s):  
Multicriteria Optimization ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Solution Set ◽  
Mixed Design ◽  
Design Variables ◽  
Definition Of ◽  
Mixed Design Variables

Properties of the Solution Set for Generalized Equilibrium Problems with Lower and Upper Bounds in FC-Metric Spaces

2013 ◽  
Vol 671-674 ◽  
pp. 1557-1560
Author(s):  
Kai Ting Wen
Keyword(s):  
Metric Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Solution Set ◽  
Lower And Upper Bounds ◽  
Paper Properties ◽  
Special Cases ◽  
Mathematical Problems ◽  
Generalized Equilibrium

The equilibrium problem includes many fundamental mathematical problems, e.g., optimization problems, saddle point problems, fixed point problems, economics problems, comple- mentarity problems, variational inequality problems, mechanics, engineering, and others as special cases. In this paper, properties of the solution set for generalized equilibrium problems with lower and upper bounds in FC-metric spaces are studied. In noncompact setting, we obtain that the solution set for generalized equilibrium problems with lower and upper bounds is nonempty and compact. Our results improve and generalize some recent results in the reference therein.

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