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Single-Loop Deterministic and Stochastic Interior-Point Algorithms for Nonlinearly Constrained Optimization

Frank E. Curtis, Xin Jiang, Qi Wang

Abstract

An interior-point algorithm framework is proposed, analyzed, and tested for solving nonlinearly constrained continuous optimization problems. The main setting of interest is when the objective and constraint functions may be nonlinear and/or nonconvex, and when constraint values and derivatives are tractable to compute, but objective function values and derivatives can only be estimated. The algorithm is intended primarily for a setting that is similar for stochastic-gradient methods for unconstrained optimization, namely, the setting when stochastic-gradient estimates are available and employed in place of gradients of the objective, and when no objective function values (nor estimates of them) are employed. This is achieved by the interior-point framework having a single-loop structure rather than the nested-loop structure that is typical of contemporary interior-point methods. For completeness, convergence guarantees for the framework are provided both for deterministic and stochastic settings. Numerical experiments show that the algorithm yields good performance on a large set of test problems.

Single-Loop Deterministic and Stochastic Interior-Point Algorithms for Nonlinearly Constrained Optimization

Abstract

An interior-point algorithm framework is proposed, analyzed, and tested for solving nonlinearly constrained continuous optimization problems. The main setting of interest is when the objective and constraint functions may be nonlinear and/or nonconvex, and when constraint values and derivatives are tractable to compute, but objective function values and derivatives can only be estimated. The algorithm is intended primarily for a setting that is similar for stochastic-gradient methods for unconstrained optimization, namely, the setting when stochastic-gradient estimates are available and employed in place of gradients of the objective, and when no objective function values (nor estimates of them) are employed. This is achieved by the interior-point framework having a single-loop structure rather than the nested-loop structure that is typical of contemporary interior-point methods. For completeness, convergence guarantees for the framework are provided both for deterministic and stochastic settings. Numerical experiments show that the algorithm yields good performance on a large set of test problems.
Paper Structure (13 sections, 13 theorems, 84 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 13 sections, 13 theorems, 84 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Limitations
  3. Notation
  4. Outline
  5. Problem Formulation and Preliminaries
  6. Algorithm
  7. Analysis
  8. Deterministic Setting
  9. Stochastic Setting
  10. Discussion of Assumptions \ref{['ass.c']} and \ref{['ass.H']}
  11. Numerical Experiments
  12. Conclusion
  13. Appendix: Algorithm for Finding a Strictly Feasible Point

Key Result

Lemma 3.1

For all $k \in \mathbb{N}^{}$ generated in any run of Algorithm alg.slip, it follows that the iterate satisfies $x_k \in {\cal E} \cap {\cal N}(\theta_{k-1}) \subseteq {\cal F}_{<0}$ and the search direction satifies $d_k \in \mathop{\mathrm{Null}}\nolimits(A)$.

Figures (2)

  • Figure 1: On the left, an illustration of Example \ref{['ex.2.dimensional']} for which Assumption \ref{['ass.c']} holds as long as the barrier parameter $\mu$ is sufficiently large relative to $\theta$ (recall that $\mu_k/\theta_{k-1}$ is constant in the algorithm) amongst other parameter choices. Since $\mu$ is sufficiently large relative to $\theta$, it follows that a direction $d$ pointing into the interior of the polar cone of nearly active constraint gradients is also one of sufficient decrease for the barrier-augmented objective function. On the right, an illustration of Example \ref{['ex.2.dim.failure']} for which Assumption \ref{['ass.c']} fails to hold since, as $x$ approaches the origin from within the feasible region, there is no minimum value for the ratio $\mu/\theta$ such that a direction into the interior of the polar cone of nearly active constraint gradients satisfies \ref{['eq.d_conditions']}.
  • Figure 2: Histogram of relative stationarity values for experiments with a deterministic version of SLIP over problems from the CUTEst collection with a strictly feasible initial point.

Theorems & Definitions (26)

  • Lemma 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Theorem 4.1
  • ...and 16 more