Explicit continuation methods with L-BFGS updating formulas for linearly constrained optimization problems

TL;DR

This paper introduces an explicit continuation method with L-BFGS updates for linearly constrained optimization, significantly reducing computational time and memory usage compared to traditional methods like SQP and Ptctr.

Contribution

The paper proposes a new explicit continuation method (Eptctr) that is more efficient in computation and memory, with proven global convergence for linearly constrained problems.

Findings

Eptctr reduces computational time to about one-tenth of Ptctr.

Eptctr consumes only 0.4% of SQP's time in numerical tests.

Eptctr requires about one-fifth of SQP's memory.

Abstract

This paper considers an explicit continuation method with the trusty time-stepping scheme and the limited-memory BFGS (L-BFGS) updating formula (Eptctr) for the linearly constrained optimization problem. At every iteration, Eptctr only involves three pairs of the inner product of vector and one matrix-vector product, other than the traditional and representative optimization method such as the sequential quadratic programming (SQP) or the latest continuation method such as Ptctr \cite{LLS2020}, which needs to solve a quadratic programming subproblem (SQP) or a linear system of equations (Ptctr). Thus, Eptctr can save much more computational time than SQP or Ptctr. Numerical results also show that the consumed time of EPtctr is about one tenth of that of Ptctr or one fifteenth to 0.4 percent of that of SQP. Furthermore, Eptctr can save the storage space of an $(n+m) \times (n+m)$ large-scale matrix, in comparison to SQP. The required memory of Eptctr is about one fifth of that of SQP. Finally, we also give the global convergence analysis of the new method under the standard assumptions.