v2010.10.26 - Convex Optimization

216 CHAPTER 2. CONVEX GEOMETRYa point x from its coordinates t ⋆ (x) via (489), but does not prescribe theindex set I . There are at least two computational methods for specifying{Γj(i) ∗ } : one is combinatorial but sure to succeed, the other is a geometricmethod that searches for a minimum of a nonconvex function. We describethe latter:Convex problem (P)(P)maximize tt∈Rsubject to x − tv ∈ Kminimize λ T xλ∈R nsubject to λ T v = 1λ ∈ K ∗ (D) (490)is equivalent to definition (486) whereas convex problem (D) is its dual; 2.88meaning, primal and dual optimal objectives are equal t ⋆ = λ ⋆T x assumingSlater’s condition (p.285) is satisfied. Under this assumption of strongduality, λ ⋆T (x − t ⋆ v)= t ⋆ (1 − λ ⋆T v)=0; which implies, the primal problemis equivalent tominimize λ ⋆T (x − tv)t∈R(P) (491)subject to x − tv ∈ Kwhile the dual problem is equivalent tominimize λ T (x − t ⋆ v)λ∈R nsubject to λ T v = 1 (D) (492)λ ∈ K ∗Instead given coordinates t ⋆ (x) and a description of cone K , we proposeinversion by alternating solution of primal and dual problemsN∑minimize Γ ∗Tx∈R n i (x − t ⋆ iΓ i )i=1(493)subject to x − t ⋆ iΓ i ∈ K , i=1... N2.88 Form a Lagrangian associated with primal problem (P):L(t, λ) = t + λ T (x − tv) = λ T x + t(1 − λ T v) ,λ ≽K ∗ 0suptL(t, λ) = λ T x , 1 − λ T v = 0Dual variable (Lagrange multiplier [250, p.216]) λ generally has a nonnegative sense forprimal maximization with any cone membership constraint, whereas λ would have anonpositive sense were the primal instead a minimization problem having a membershipconstraint.