Gradient of the Value Function in Parametric Convex Optimization Problems

TL;DR

This paper derives a general expression for the gradient of the value function in parametric convex optimization problems, highlighting conditions for differentiability in quadratic programs.

Contribution

It provides a unified formula for the gradient of the value function and establishes differentiability conditions for quadratic programs under certain constraint qualifications.

Findings

Gradient of the value function expressed in terms of problem data

Differentiability of the value function in quadratic programs

Conditions for continuous differentiability in feasible regions

Abstract

We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange multipliers. In particular, we show that for the strictly convex parametric quadratic program the value function is continuously differentiable at every point in the interior of feasible space for which the Linear Independent Constraint Qualification holds.