Convex Optimization · Problem 4.49

Duality — Boyd & Vandenberghe

Mathematics — Optimization · generalized-fractional-programming

Problem

Generalized fractional programming. Let

K \in \mathbf{R}^m

be a proper cone. Show that the function

f_0 : \mathbf{R}^n \to \mathbf{R}^m

, defined by

f_0(x) = \inf\{t \mid Cx + d \preceq_K t(Fx + g)\}, \qquad \mathbf{dom}\, f_0 = \{x \mid Fx + g \succ_K 0\},

with

C, F \in \mathbf{R}^{m \times n}

d, g \in \mathbf{R}^m

, is quasiconvex. A quasiconvex optimization problem with objective function of this form is called a generalized fractional program. Express the generalized linear-fractional program of page 152 and the generalized eigenvalue minimization problem (4.73) as generalized fractional programs.

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