Convex Optimization · Problem 6.11

Statistical estimation — Boyd & Vandenberghe

Mathematics — Optimization · least-squares-direction-interpolation

Problem

Least-squares direction interpolation. Suppose

F_1, \ldots, F_n : \mathbf{R}^k \to \mathbf{R}^p

, and we form the linear combination

F : \mathbf{R}^k \to \mathbf{R}^p

F(u) = x_1 F_1(u) + \cdots + x_n F_n(u),

where

x

is the variable in the interpolation problem. In this problem we require that

\angle(F(v_j), q_j) = 0

j = 1, \ldots, m

, where

q_j

are given vectors in

\mathbf{R}^p

, which we assume satisfy

\|q_j\|_2 = 1

. In other words, we require the direction of

F

to take on specified values at the points

v_j

. To ensure that

F(v_j)

is not zero (which makes the angle undefined), we impose the minimum length constraints

\|F(v_j)\|_2 \geq \epsilon

j = 1, \ldots, m

, where

\epsilon > 0

is given. Show how to find

x

that minimizes

\|x\|^2

, and satisfies the direction (and minimum length) conditions above, using convex optimization.