Convex Optimization 2026, 20 - Generalized Inequalities

A problem with generalized inequality constraints with K_i ⊆ ℝ^k_i proper cones, and not assuming convexity of the problem, along with 𝔻 = ⋂^m_i=0 dom f_i ∩ ⋂^p_i=1 dom h_i ≠ ∅. With each generalized inequality f_i(x) ⪯_Ki 0, associate a Lagrange multiplier λ_i ∈ ℝ^k_i and the classically defined (full) Lagrangian. Its dual function is as in a problem with scalar inequalities (p. 264). In a problem with scalar inequalities, the dual function gives lower bounds on the optimal value of the primal problem. After extension of non-negativity onto the dual cone of K_i (i.e. dual nonnegativity), weak duality follows immediately (5.9.1.0). Strong duality holds when the primal problem is convex and satisfies an appropriate constraint qualification. A generalized version of Slater's condition for the problem is that ∃ x ∈ relint 𝔻 with Ax = b; f_i(x) ≺_Ki 0. This implies strong duality and the existence of a dual optimum (5.9.1.1).

The optimality conditions can be extended to problems with generalized inequalities. Complementary slackness assumes primarily that primal and dual optimal values are equal, and optimal points are attained. The complementary slackness conditions follow directly from equality f₀(x*) = g(λ*, ν*). By decomposing the definition of g, gain λ_i*^Tf_i(x*) = 0, which generalizes complementary slackness. Similarly, KKT is acquired through the assumption that f_i, h_i are differentiable (5.9.2). With similar methods, generalize perturbation and sensitivity analysis and theorems of alternatives.