Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An objective-function-free algorithm for nonconvex stochastic optimization with deterministic equality and inequality constraints

S. Gratton, Ph. L. Toint

Abstract

An algorithm is proposed for solving optimization problems with stochastic objective and deterministic equality and inequality constraints. This algorithm is objective-function-free in the sense that it only uses the objective's gradient and never evaluates the function value. It is based on an adaptive selection of function-decreasing and constraint-improving iterations, the first ones using an Adagrad-type stepsize. When applied to problems with full-rank Jacobian, the combined primal-dual optimality measure is shown to decrease at the rate of O(1/sqrt{k}), which is identical to the convergence rate of first-order methods in the unconstrained case.

An objective-function-free algorithm for nonconvex stochastic optimization with deterministic equality and inequality constraints

Abstract

An algorithm is proposed for solving optimization problems with stochastic objective and deterministic equality and inequality constraints. This algorithm is objective-function-free in the sense that it only uses the objective's gradient and never evaluates the function value. It is based on an adaptive selection of function-decreasing and constraint-improving iterations, the first ones using an Adagrad-type stepsize. When applied to problems with full-rank Jacobian, the combined primal-dual optimality measure is shown to decrease at the rate of O(1/sqrt{k}), which is identical to the convergence rate of first-order methods in the unconstrained case.

Paper Structure

This paper contains 12 sections, 12 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An algorithm
  3. Complexity analysis
  4. Assumptions
  5. Normal steps
  6. Tangential steps
  7. Telescoping sum for values of the Lyapunov function
  8. Tangential complexity
  9. Normal complexity
  10. Combined complexity
  11. Further discussion of the stochastic conditions
  12. Summary and open questions

Key Result

Lemma 2.1

Suppose that a tangential step $s_{T,k}$ is computed at iteration $k$. Then and

Theorems & Definitions (12)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 2 more