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Any concave (strictly concave) function is quasi-concave (strictly quasi-concave), but the converse is not true. |+ht. @H      q(fs pef[DU_12 bQQpeA function f defined on a convex set Xn is said to be quasi-convx, if for any x and x in X and any l[0, 1]: f((1-l)x +lx ) min {f(x ),f(x )} f is said to be strictly quasi-convex, if the sign   in the above inequality is replaced with  < . . Any convex (strictly convex) function is quasi-convex (strictly quasi-convex), but the converse is not true. {Z,ZhZrZZ/ >H        =-)fq pef[DU_14 I{ 0. A necessary and sufficient condition for a function f defined on a convex set Xn to be quasi-concave is that all the superior sets S(y) are convex; f is strictly quasi-concave if all the superior sets S(y) are convex, and for any two points x and x in any superior set S(y), the points on the line segment {x=(1-l)x +lx : l[0, 1]} expect possibly the two endpoints are all contained in Int(S(y)).tZZL       P"*"#"#"#I"lS $!pef[DU_16 QĉR: Assume that f and g are differentiable functions defined on a convex set Xn and non-decreasing in each variable. Assume that f is quasi-concave and g is quasi-convex, and that f(O)=g(O)=0. Then both of the primal and the dual convex programming problems (where c>0 and k>0 are constants): max f(x), S.T. g(x)<c, min g(x), S.T. f(x)>k, have optimal solutions. The solution of the primal (resp.the dual) problem is a tangent point of g(x)=c (resp. f(x)=k) with a level set of f (resp. g). +#"K"*" """""jFS1!  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