Abstract
We present a combined homotopy interior‐point method for a general multiobjective programming problem. For solving the KKT points of the multiobjective programming problem, the homotopy equation is constructed. We prove the existence and convergence of a smooth homotopy path from almost any initial interior point to a solution of the KKT system under some basic assumptions.
Highlights
IntroductionWe consider the following multiobjective programming problem: min f x , s.t. g x ≤ 0, h x 0, MOP where f f1, f2, . . . , fp T : Rn → Rp, g h1, h2, . . . , hs T : Rn → Rs
We consider the following multiobjective programming problem: min f x, s.t. g x ≤ 0, h x 0, MOP where f f1, f2, . . . , fp T : Rn → Rp, g h1, h2, . . . , hs T : Rn → Rs.g1, g2, . . . , gm T : Rn → Rm and hSince Kellogg et al 1 and Smale 2 have published their remarkable papers, more and more attention has been paid to the homotopy method
We present a combined homotopy interior-point method for a general multiobjective programming problem
Summary
We consider the following multiobjective programming problem: min f x , s.t. g x ≤ 0, h x 0, MOP where f f1, f2, . . . , fp T : Rn → Rp, g h1, h2, . . . , hs T : Rn → Rs. Song and Yao 10 generalized the combined homotopy interior-point method to the general multiobjective programming problem under the so-called normal cone condition In that paper, they proved the existence of the homotopy path under the following assumptions: A1 Ω0 is nonempty and bounded; A2 for all x ∈ Ω, the vectors {∇gi x , i ∈ B x , ∇hj x , j ∈ J} are linearly independent; A3 for all x ∈ Ω, {x B x } Ω {x}, i∈B x ui∇gi x j∈J zj ∇hj x : z zj ∈ Rs, ui ≥ 0, i ∈. The purpose of this paper is to generalize the combined homotopy interior-point method for a general multiobjective programming problem MOP under quasinorm cone condition that weakens the assumptions more than the ones in 10 and constructs a new homotopy equation which is much different and simpler the that one given in 9.
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