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A Sequential Cubic Programming Method with Second-Order Complexity Guarantees for Equality Constrained Optimization

Nikos Dimou, Michael J. O'Neill

Abstract

We develop a new method for equality constrained optimization problems based on a sequential cubic programming framework. Each iteration utilizes a step decomposition based on the Jacobian of the constraints into a normal and a tangential component, the latter of which is found by solving a subproblem involving cubic regularization. The method incorporates second-order correction steps as necessary to ensure global convergence to second-order stationary points as well as local quadratic convergence. In addition, we show that the algorithm is the first to obtain worst case complexity guarantees on the order of $\mathcal{O}(ε_g^{-3/2})$ for the gradient of the Lagrangian, $\mathcal{O}(ε_H^{-3})$ in terms of second-order stationarity, and $\mathcal{O}(ε_c^{-1})$ in terms of the constraint violation. These are the best known complexity guarantees of any method for this class of problems.

A Sequential Cubic Programming Method with Second-Order Complexity Guarantees for Equality Constrained Optimization

Abstract

We develop a new method for equality constrained optimization problems based on a sequential cubic programming framework. Each iteration utilizes a step decomposition based on the Jacobian of the constraints into a normal and a tangential component, the latter of which is found by solving a subproblem involving cubic regularization. The method incorporates second-order correction steps as necessary to ensure global convergence to second-order stationary points as well as local quadratic convergence. In addition, we show that the algorithm is the first to obtain worst case complexity guarantees on the order of for the gradient of the Lagrangian, in terms of second-order stationarity, and in terms of the constraint violation. These are the best known complexity guarantees of any method for this class of problems.

Paper Structure

This paper contains 22 sections, 35 theorems, 184 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Organization
  5. Notation
  6. Sequential Cubic Programming Method
  7. The Lagrangian multipliers
  8. Merit function
  9. The Maratos effect and Negative Curvature
  10. The SCP Algorithm
  11. First-order stationary points
  12. First-order Convergence
  13. First-order Complexity
  14. Second-order stationary points
  15. Second-order Convergence
  16. ...and 7 more sections

Key Result

Lemma 2.1

Suppose $u^*_k$ is an optimal solution of eq:mku. Then, $u_k^*$ satisfies or1-or3. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (69)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • ...and 59 more