Approximate solutions of vector optimization problems via improvement sets in real linear spaces
We deal with a constrained vector optimization problem between real linear spaces without assuming any topology and by considering an ordering defined through an improvement set E. We study E-optimal and weak E-optimal solutions and also proper E-optimal solutions in the senses of Benson and Henig. We relate these types of solutions and we characterize them through approximate solutions of scalar optimization problems via linear scalarizations and nearly E-subconvexlikeness assumptions. Moreover, in the particular case when the feasible set is defined by a cone-constraint, we obtain characterizations by means of Lagrange multiplier rules. The use of improvement sets allows us to unify and to extend several notions and results of the literature. Illustrative examples are also given.
KeywordsVector optimization Improvement set Approximate weak efficiency Approximate proper efficiency Nearly E-subconvexlikeness Linear scalarization Lagrange multipliers algebraic interior Vector closure
Mathematics Subject ClassificationPrimary 90C26 90C29 Secondary 90C46 90C48 49K27
The authors are grateful to the anonymous referee and the Associated Editor for their useful suggestions and remarks.
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