数学附录 15二元凹函数和拟凹函数的判别: Assume that f( x,y is defined on a convex set X and fis C2. Then f is concave if f <0 and f XXy -(fx)2>0 A necessary and sufficient condition for a function f defined on a convex set crn 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 xand x in any superior set S(y), the points on the line segment x=(1-mx+x: nEIO, I expect possibly the two endpoints are all contained in Int(s(y))
数学附录 • 15 二元凹函数和拟凹函数的判别： • Assume that f(x,y) is defined on a convex set X and f is C2 . Then f is concave if f xx < 0 and f xx fyy –(f xy ) 2 > 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-)x’+x”: [0, 1]} expect possibly the two endpoints are all contained in Int(S(y))
数学附录 16凸规划: Assume that f and g are differentiable functions defined on a convex set XoRn and non- decreasing in each variable. Assume that f is quasi concave and g is quasi-convex, and that f(O=gO=0. Then both of the primal and the dual convex programming problems (where c> and kare constants): maxf(x),S.T.g(x)≤c, min g(X), s.T. f(x)2k, 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)
数学附录 • 16 凸规划: Assume that f and g are differentiable functions defined on a convex set Xn and nondecreasing in each variable. Assume that f is quasiconcave 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)