Abstract
Abstract In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions g i {g}_{i} are nonsmooth on Banach space E, we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.
Highlights
In the last few decades, the methodology for solving optimization problems has been widely applied to many research fields
It is well known that the Karush-Kuhn-Tucker (KKT) optimality conditions play an important role in the study of optimization theory
If the objective functions of optimization problems are taken as real numbers, they are categorized as deterministic optimization problems
Summary
In the last few decades, the methodology for solving optimization problems has been widely applied to many research fields (see [1,2,3,4,5,6,7,8,9,10,11,12,13]). We shall investigate the following interval-valued robust optimization problem (IVROP):. If the constraint functions gi are nonlinear and nonsmooth on the Banach space E for each i ∈ {1, 2, ..., m}, we shall establish the robust KKT necessary optimality conditions. For each i = 1, 2, ..., m, if constraint functions are independent of vi, problem (1.1) reduces to the following interval-valued optimization problem: inf f (x) = [ f L(x), f U (x)] (IVOP) subject to K = {x ∈ E | gi(x) ≤ 0, i = 1, 2, ..., m}. Under the assumptions that each gi(x) is convex and continuously differentiable, Ishibuchi and Tanaka in [36], and Inuiguchi and Kume in [37], and Wu in [38,39] studied the KKT necessary optimality conditions of the interval-valued optimization
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