scholarly journals Existence of Pareto-optimal solutions to the vector optimization problem with an unbounded feasible set

2015 ◽  
pp. 27-31 ◽  
Author(s):  
T.I. Sergienko ◽  
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Feasible Set

scholarly journals Evaluation Method of Multiobjective Functions’ Combination and Its Application in Hydrological Model Evaluation

2020 ◽  
Vol 2020 ◽  
pp. 1-23 ◽  
Author(s):  
Jiuyuan Huo ◽  
Liqun Liu
Keyword(s):  
Objective Function ◽  
Parameter Optimization ◽  
Optimization Problem ◽  
Hydrological Model ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Hydrological Forecasting ◽  
Model Parameter Optimization

Parameter optimization of a hydrological model is intrinsically a high dimensional, nonlinear, multivariable, combinatorial optimization problem which involves a set of different objectives. Currently, the assessment of optimization results for the hydrological model is usually made through calculations and comparisons of objective function values of simulated and observed variables. Thus, the proper selection of objective functions’ combination for model parameter optimization has an important impact on the hydrological forecasting. There exist various objective functions, and how to analyze and evaluate the objective function combinations for selecting the optimal parameters has not been studied in depth. Therefore, to select the proper objective function combination which can balance the trade-off among various design objectives and achieve the overall best benefit, a simple and convenient framework for the comparison of the influence of different objective function combinations on the optimization results is urgently needed. In this paper, various objective functions related to parameters optimization of hydrological models were collected from the literature and constructed to nine combinations. Then, a selection and evaluation framework of objective functions is proposed for hydrological model parameter optimization, in which a multiobjective artificial bee colony algorithm named RMOABC is employed to optimize the hydrological model and obtain the Pareto optimal solutions. The parameter optimization problem of the Xinanjiang hydrological model was taken as the application case for long-term runoff prediction in the Heihe River basin. Finally, the technique for order preference by similarity to ideal solution (TOPSIS) based on the entropy theory is adapted to sort the Pareto optimal solutions to compare these combinations of objective functions and obtain the comprehensive optimal objective functions’ combination. The experiments results demonstrate that the combination 2 of objective functions can provide more comprehensive and reliable dominant options (i.e., parameter sets) for practical hydrological forecasting in the study area. The entropy-based method has been proved that it is effective to analyze and evaluate the performance of different combinations of objective functions and can provide more comprehensive and impersonal decision support for hydrological forecasting.

scholarly journals Lipschitz and differentiable dependence of solutions on a parameter in a scalarization method

1987 ◽  
Vol 42 (3) ◽  
pp. 353-364 ◽  
Author(s):  
Alicia Sterna-Karwat
Keyword(s):  
Vector Optimization ◽  
Normed Space ◽  
Convex Cone ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Optimal Solutions ◽  
Scalar Problem ◽  
Lipschitz Continuous ◽  
Scalarization Method ◽  
Convexity Assumptions

AbstractThis paper is concerned with a vector optimization problem set in a normed space where optimality is defined through a convex cone. The vector problem can be solved using a parametrized scalar problem. Under some convexity assumptions, it is shown that dependence of optimal solutions on the parameter is Lipschitz continuous. Hence differentiable dependence on the solutions on the parameter is derived.

New Dynamic Multi-Objective Constrained Optimization Evolutionary Algorithm

2015 ◽  
Vol 32 (05) ◽  
pp. 1550036 ◽  
Author(s):  
Chun-An Liu ◽  
Yuping Wang ◽  
Aihong Ren
Keyword(s):  
Constrained Optimization ◽  
Evolutionary Algorithm ◽  
Optimization Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Constrained Optimization Problem ◽  
Multi Objective ◽  
Time Period ◽  
Pareto Fronts

For dynamic multi-objective constrained optimization problem (DMCOP), it is important to find a sufficient number of uniformly distributed and representative dynamic Pareto optimal solutions. In this paper, the time period of the DMCOP is first divided into several random subperiods. In each random subperiod, the DMCOP is approximately regarded as a static optimization problem by taking the time subperiod fixed. Then, in order to decrease the amount of computation and improve the effectiveness of the algorithm, the dynamic multi-objective constrained optimization problem is further transformed into a dynamic bi-objective constrained optimization problem based on the dynamic mean rank variance and dynamic mean density variance of the evolution population. The evolution operators and a self-check operator which can automatically checkout the change of time parameter are introduced to solve the optimization problem efficiently. And finally, a dynamic multi-objective constrained optimization evolutionary algorithm is proposed. Also, the convergence analysis for the proposed algorithm is given. The computer simulations are made on four dynamic multi-objective optimization test functions and the results demonstrate that the proposed algorithm can effectively track and find the varying Pareto optimal solutions or the varying Pareto fronts with the change of time.

scholarly journals Sensitivity analysis for parametric vector optimization problems using differential equations approach

2001 ◽  
Vol 25 (9) ◽  
pp. 621-628
Author(s):  
Fatma M. Ali
Keyword(s):  
Differential Equations ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Parametric Vector Optimization ◽  
Local Solutions

A new method for obtaining sensitivity information for parametric vector optimization problems(VOP)vis presented, where the parameters in the objective functions and anywhere in the constraints. This method depends on using differential equations technique for solving multiobjective nonlinear programing problems which is very effective in finding many local Pareto optimal solutions. The behavior of the local solutions for slight perturbation of the parameters in the neighborhood of their chosen initial values is presented by using the technique of trajectory continuation. Finally some examples are given to show the efficiency of the proposed method.

scholarly journals Stability by the vector criterion of a mixed integer optimization problem with quadratic criterial fun ctions

2020 ◽  
pp. 15-21
Author(s):  
Т.Т. Lebedeva ◽  
◽  
N.V. Semenova ◽  
T.I. Sergienko ◽  
◽  
...  
Keyword(s):  
Initial Data ◽  
Optimization Problem ◽  
Sufficient Conditions ◽  
Mixed Integer ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Integer Optimization ◽  
Mixed Integer Optimization ◽  
Vector Criterion

The article is devoted to the study of qualitative characteristics of different concepts of stability of vector problems of mixed-integer optimization, namely, to identifying the conditions under which the set of Pareto-optimal solutions of the problem possesses some property of invariance defined in advance in relation to the external influences on initial data of the problem. We investigate the questions of stability with respect to data perturbations in a vector criterion of mixed-integer optimization problem. The necessary and sufficient conditions of stability of three types for a problem of finding the solutions of the Pareto set are found. Such conditions guarantee that the small variations of initial data of vector criterion: 1) do not result in new Paretooptimal solutions, 2) save all Pareto-optimal solutions of the problem and can admit new solutions, 3) do not change the set of Pareto-optimal solutions of the initial problem.

scholarly journals Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Paolo Bevilacqua ◽  
Gianni Bosi ◽  
Magalì Zuanon
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problem ◽  
Design Space ◽  
Pareto Optimal ◽  
Multiobjective Optimization Problem ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Maximal Elements ◽  
Compact Design

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.

Multiobjective Hybrid Optimization–Antioptimization for Force Design of Tensegrity Structures

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
Makoto Ohsaki ◽  
Jingyao Zhang ◽  
Isaac Elishakoff
Keyword(s):  
Optimization Problem ◽  
Hybrid Approach ◽  
Programming Approach ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Tensegrity Structures ◽  
Tangent Stiffness Matrix ◽  
Lowest Eigenvalue

Properties of Pareto optimal solutions considering bounded uncertainty are first investigated using an illustrative example of a simple truss. It is shown that the nominal values of the Pareto optimal solutions considering uncertainty are slightly different from those without considering uncertainty. Next a hybrid approach of multiobjective optimization and antioptimization is presented for force design of tensegrity structures. We maximize the lowest eigenvalue of the tangent stiffness matrix and minimize the deviation of forces from the specified target distribution. These objective functions are defined as the worst values due to the possible errors in the fabrication and construction processes. The Pareto optimal solutions are found by solving the two-level optimization–antioptimization problems using a nonlinear programming approach for the upper optimization problem and enumeration of the vertices of the uncertain region for the lower antioptimization problem.

scholarly journals Optimality and duality in nonsmooth vector optimization with non-convex feasible set

2020 ◽  
Author(s):  
Dr. Sunila Sharma ◽  
Priyanka Yadav
Keyword(s):  
Objective Function ◽  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Convex Programming Problem ◽  
Feasible Set ◽  
Nonsmooth Vector Optimization ◽  
Necessary And Sufficient

For a convex programming problem, the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for optimality under suitable constraint qualification. Recently, Suneja et al proved KKT optimality conditions for a differentiable vector optimization problem over cones in which they replaced the cone-convexity of constraint function by convexity of feasible set and assumed the objective function to be cone-pseudoconvex. In this paper, we have considered a nonsmooth vector optimization problem over cones and proved KKT type sufficient optimality conditions by replacing convexity of feasible set with the weaker condition considered by Ho and assuming the objective function to be generalized nonsmooth cone-pseudoconvex. Also, a Mond-Weir type dual is formulated and various duality results are established in the modified setting.

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