Theorem 12.1: Consider the following convex programming problem

minψqr(ψ)Subject tohi(ψ)0,i=1,,m

where r and hi, i = 1, …, m are convex and there exists a ψ′ such that hi(ψ′) > 0 for all i = 1, 2, …, m. Define the Lagrangian


where θ = (θ1, ., θm)T. If ψ* solves the convex programming problem, then there exists Lagrange multipliers θ*=(θ1*,,θm*)T such that the following three conditions hold:

L(ψ*,θ)L(ψ*,θ*)L(ψ,θ*)for all ψandθ0






This theorem tells us that ψ* can be found by solving the unconstrained optimization problem minψL(ψ, θ*), and that θ* can be found by solving the Lagrangian dual problem maxθ≥0L(ψ*, θ). Although it ...

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