Optimality and Duality with Respect to <i>b</i>-(<i>ℰ</i>,<i>m</i>)-Convex Programming

• Main Authors: Bo Yu, Jiagen Liao, Tingsong Du
• Format: Article
• Language: English
• Published: MDPI AG 2018-12-01
• Series: Symmetry
• Subjects: (<i>ℰ</i>,<i>m</i>)-convex sets; <i>b</i>-(<i>ℰ</i>,<i>m</i>)-convex mappings; optimality conditions; duality theorems
• Online Access: https://www.mdpi.com/2073-8994/10/12/774

Noticing that <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">E</mi> </semantics> </math> </inline-formula>-convexity, <i>m</i>-convexity and <i>b</i>-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex sets and the <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex mappings are introduced. The properties concerning operations that preserve the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convexity of the proposed mappings are derived. The unconstrained and inequality constrained <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz&#8315;John necessary optimality criteria for nonlinear multi-objective <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex programming are established. The Wolfe-type symmetric duality theorems under the <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in <i>b</i>-<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">E</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-convex programming.