# Solvers

## LP

HiGHS has implementations of the three main solution techniques for LP. HiGHS will choose the most appropriate technique for a given problem, but this can be over-ridden by setting the option **solver**.

#### Simplex

HiGHS has efficient implementations of both the primal and dual simplex methods, although the dual simplex solver is likely to be faster and is more robust, so is used by default. The novel features of the dual simplex solver are described in

*Parallelizing the dual revised simplex method*, Q. Huangfu and J. A. J. Hall, Mathematical Programming Computation, 10 (1), 119-142, 2018 DOI: 10.1007/s12532-017-0130-5.

- Setting the option
**solver**to "simplex" forces the simplex solver to be used - The option
**simplex_strategy**determines whether the primal solver or one of the parallel solvers is to be used.

#### Interior point

HiGHS has one interior point (IPM) solver based on the preconditioned conjugate gradient method, as discussed in

*Implementation of an interior point method with basis preconditioning*, Mathematical Programming Computation, 12, 603-635, 2020. DOI: 10.1007/s12532-020-00181-8.

This solver is serial. An interior point solver based on direct factorization is being developed.

Setting the option **solver** to "ipm" forces the IPM solver to be used

#### Primal-dual hybrid gradient method

HiGHS includes the cuPDLP-C primal-dual hybrid gradient method for LP (PDLP). Currently this only runs on CPU, so it is unlikely to be competitive with the HiGHS interior point or simplex solvers. Enabling HiGHS to run PDLP on a GPU is work in progress.

Setting the option **solver** to "pdlp" forces the PDLP solver to be used

## MIP

The HiGHS MIP solver uses established branch-and-cut techniques

## QP

The HiGHS solver for convex QP problems uses an established primal active set method. The new interior point solver will also be able to solve convex QP problems.