This paper analyzes the convergence rates of the Frank-Wolfe method for solving convex constrainedmultiobjective optimization. We establish improved convergence rates under different assumptions on the objective func...
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This paper analyzes the convergence rates of the Frank-Wolfe method for solving convex constrainedmultiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. Notably, we demonstrate that the method can achieve linear convergence rates in terms of a merit function whenever the objectives are strongly convex and the limit point is in the relative interior of the feasible set, or when the feasible set is strongly convex and it does not contain an unconstrained weak Pareto point. Moreover, improved sublinear convergence rates can also be obtained in other scenarios where the feasible set is uniformly convex. Additionally, we explore enhanced convergence rates with respect to an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.
In this paper, we propose and analyze an away-step Frank-Wolfe algorithm designed for solving multiobjective optimization problems over polytopes. We prove that each limit point of the sequence generated by the algori...
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In this paper, we propose and analyze an away-step Frank-Wolfe algorithm designed for solving multiobjective optimization problems over polytopes. We prove that each limit point of the sequence generated by the algorithm is a weak Pareto optimal solution. Furthermore, under additional conditions, we show linear convergence of the whole sequence to a Pareto optimal solution. Numerical examples illustrate a promising performance of the proposed algorithm in problems where the multiobjective Frank-Wolfe convergence rate is only sublinear.
The optimization of a microgrid is a typical multivariable nonlinear programming problem with multiple hybrid constraints. In this paper, a kind of constrainedmultiobjective optimization based on nondominated immune ...
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The optimization of a microgrid is a typical multivariable nonlinear programming problem with multiple hybrid constraints. In this paper, a kind of constrainedmultiobjective optimization based on nondominated immune algorithm is used, in which the constraint is transformed into a target function, a relatively isolated nondominated antibody is selected as the active antibody, and ratio cloning, recombination, and mutation are performed in order, according to the crowding distance of the active antibody, thus obtaining the convergence of an ideal Pareto front and a uniform distribution of the Pareto-optimal solution. The optimization model of the power output of the microgrid and the simulation of the operation in two microgrids for 24 h are carried out by utilizing the algorithm. The results show that good environmental protection is obtained at the lowest cost of running and maintenance, thus satisfying the stability of power supply requirements for users. (c) 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.
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