By S.M. Stefanov
In this publication, the writer considers separable programming and, particularly, considered one of its very important instances - convex separable programming. a few normal effects are provided, thoughts of approximating the separable challenge by way of linear programming and dynamic programming are thought of.
Convex separable courses topic to inequality/ equality constraint(s) and boundaries on variables also are studied and iterative algorithms of polynomial complexity are proposed.
As an program, those algorithms are utilized in the implementation of stochastic quasigradient easy methods to a few separable stochastic courses. Numerical approximation with admire to I1 and I4 norms, as a convex separable nonsmooth unconstrained minimization challenge, is taken into account to boot.
Audience: complicated undergraduate and graduate scholars, mathematical programming/ operations examine specialists.
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24 (Directional differentiability of a convex function) Let X be a convex set in 1R n and f : X - t 1R be a convex function. Then the directional derivative M(xo) of f at Xo Eint X in the direction d i= 0, d E 1R n exists. Since Xo Eint X, then there exists a AO A E (0, AO) we have Xo + Ad E X. Consider the difference quotient Proof. q(A) ~f f(xo + Ad) - > 0 such that for f(x o). A Let 0 < A2 < Al :S AO' Hence 0 < ~ < 1. Since f is convex, whence f(xo + A2d ) - f(xo) < f(xo + Ald) - f(xo) A2 Al for Al > A2 > O.
N. Therefore n f(x) - f(x) == L n fj(xj) - L j=l n fJ(Xj) == j=l L [Jj(Xj) - fj(xj)] j==l n ~ L [Ajff(Xj) + (1- Aj)fT(Xj)](Xj - Xj) j=l == (f(x) , x - x). 1 Since convex functions have derivatives on the right and on the left at each interior feasible point, then the assumption that ff (Xj) and fT (Xj) exist is reasonable. 35 (Subgradient of a function in two variables) Let f(x,y) be a convexfunction ofx for each y, let there exist a y(x) such that f(x) ~f max f(x,y) = f(x,y(x)) yEY and let the subgradient h(x, y) of f(x, y) with respect to x be known for each y.
18 J is said to be a proper Junction if J(x) < 00 far at least one x E Rn and J(x) > -00 for all x ERn, or, in other words, if dom J is a nonempty set on which J(x) > -00. Otherwise J is called improper. 18 (Necessary and sufficient condition for convexity of the epigraph) Let X be a nonempty convex set in Rn and J : X -+ R. Then J is convex iJ and only iJ epi J is a convex set in R n +l . Proof. Necessity. Let J be a convex function and let (Xl, rl), (X2' r2) E epi J. Therefore J(XI) :S rl, J(X2):S r2, Xl, X2 E X.