Convex Optimization · Problem 4.61

Duality — Boyd & Vandenberghe

Mathematics — Optimization · optimization-logistic-model

Problem

Optimization with logistic model. A random variable

X \in \{0, 1\}

satisfies

\mathbf{prob}(X = 1) = p = \frac{\exp(a^T x + b)}{1 + \exp(a^T x + b)},

where

x \in \mathbf{R}^n

is a vector of variables that affect the probability, and

a

and

b

are known parameters. The variable

x

, which we are to optimize over, is subject to a set of linear constraints,

Fx \preceq g

. Formulate the following problems as convex optimization problems. (a) Maximizing buying probability. The goal is to choose

x

to maximize

p

. (b) Maximizing expected profit. Let

c^T x + d

be the profit derived from selling the product, which we assume is positive for all feasible

x

. The goal is to maximize the expected profit, which is

p(c^T x + d)