Convex optimization

In mathematics and optimization theory, a typical convex optimization problem is to minimize f(x), where f is a convex function, and x is point, of a vector space X, subject to

g₁(x) = 0,...,g_m(x) = 0,

where the g_i(x) are convex functions.

The Lagrange function for the problem (see Lagrange multipliers for the smooth function case) is

L = l₀f(x) + l₁g₁(x) + ... + l_mgm(x).

See Kuhn-Tucker theorem. Then we have

1. L(x_opt,l0_opt,l1_opt,...,lm_opt) = min_x L(l0_opt,l1_opt,...,lm_opt),
2. l0_opt ≥ 0,l1_opt ≥ 0,...,lm_opt ≥ 0
3. l_1,optg₁(x_opt) = 0, ... , l_m,optg_m(x_opt) = 0

If l_0,opt ≠ 0, so 1)-3) enough to find x_opt.

l_0,opt ≠ 0, if there exists x, so that

g1(x) < 0,...,gm(x) < 0.