Complexity certifications of first order inexact Lagrangian methods for general convex programming

TL;DR

This paper establishes computational complexity bounds for inexact dual gradient methods applied to general convex problems, with practical implementation details and an open-source solver for quadratic programs.

Contribution

It provides the first complexity certifications for inexact dual methods in convex optimization, including implementation of a new solver, DuQuad.

Findings

Sublinear complexity estimates for primal suboptimality and feasibility.

Analysis of primal last iterate and average primal iterate approaches.

Open-source solver DuQuad for quadratic programming.

Abstract

In this chapter we derive computational complexity certifications of first order inexact dual methods for solving general smooth constrained convex problems which can arise in real-time applications, such as model predictive control. When it is difficult to project on the primal constraint set described by a collection of general convex functions, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply dual (fast) gradient algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is based on two types of approximate primal solutions: the primal last iterate and an average of primal iterates. We provide sublinear computational complexity estimates on the primal suboptimality and constraint (feasibility) violation of the generated approximate primal solutions. In the final part of the chapter, we present an open-source quadratic optimization solver, referred to as DuQuad, for convex quadratic programs and for evaluation of its behaviour. The solver contains the C-language implementations of the analyzed algorithms.