强预拟不变凸函数的充要条件(英文)

The Sufficiency and Necessity Conditions of Strongly Prequasi-Invex Functions

本文引入了一类新的广义凸函数—强预拟不变凸函数.讨论了强预拟不变凸函数与预拟不变凸函数、严格预拟不变凸函数及半严格预拟不变凸函数之间的关系,得到它的三个充要条件:(i)当条件P_1满足时,f是强预拟不变凸函数的充分必要条件是f是预拟不变凸函数且f满足中间点强预拟不变凸性;(ii)当条件P_2满足时,f是强预拟不变凸函数的充分必要条件是f是严格预拟不变凸函数且f满足中间点强顶拟不变凸性;(iii)当条件P_2满足时,f是强预拟不变凸函数的充分必要条件是f是半严格预拟不变凸函数且f满足中间点强预拟不变凸性.

In this paper, a new type of generalized convex functions-strongly prequasi-invex functions is introduced. And the relationships among strongly prequasiinvex functions and prequasi-invex functions, and strictly prequasi-invex functions and semistrictly prequasi-invex functions are discussed. Three sufficiency and necessity conditions of strongly prequasi-invex functions are obtained： （i） Under Condition P1, f is a strongly prequasi-invex function if and only if f is a prequasi-invex function and f satisfies intermediate-point strong prequasi-invexity; （ii） Under Condition P2, f is a strongly prequasi-invex function if and only if f is a strictly prequasi-invex function and f satisfies intermediate-point strong prequasi-invexity; （iii） Under Condition P2, f is a strongly prequasi-invex function if and only if f is a semistrictly prequasi-invex function and f satisfies intermediate-point strong prequasi-invexity.