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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 two interior point (IPM) solvers:

  • IPX is 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.

    Setting the option solver to "ipx" forces the IPX solver to be used

  • HiPO is based on a direct factorisation, as discussed in

    A factorisation-based regularised interior point method using the augmented system, F. Zanetti and J. Gondzio, 2025, available on arxiv

    This solver is parallel.

    Setting the option solver to "hipo" forces the HiPO solver to be used.

    The hipo_system option can be used to select the approach to use when solving the Newton systems within the interior point solver: select "augmented" to force the solver to use the augmented system, "normaleq" for normal equations, or "choose" to leave the choice to the solver.

    The option hipo_ordering can be used to select the fill-reducing heuristic to use during the factorisation:

    • Nested dissection, obtained setting the option hipo_ordering to "metis".
    • Approximate mininum degree, obtained setting the option hipo_ordering to "amd".
    • Reverse Cuthill-McKee, obtained setting the option hipo_ordering to "rcm".

Setting the option solver to "ipm" selects the HiPO solver, if the build supports it, otherwise it selects the IPX solver.

Primal-dual hybrid gradient method (PDLP)

HiGHS includes the cuPDLP-C primal-dual hybrid gradient method for LP (PDLP). On Linux and Windows, this can be run on an NVIDIA GPU. On a CPU, it is unlikely to be competitive with the HiGHS interior point or simplex solvers.

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

Basic solution

The simplex solver always generates a basic solution. This is a solution at a vertex of the feasible region of the LP.

By default, the IPM solvers also generate a basic solution by running a procedure known as "crossover". This can occasionally be very expensive so if users require only a solution to the LP, and not a basic solution, crossover can be suppressed. This is done by setting the run_crossover option to "off". However, if the IPM solver terminates without satsifying the feasibility and optimality conditions, HiGHS will not claim optimality. Hence it is better to set the run\_crossover option to "choose", in which case crossover will be run if the IPM solver terminates without determining optimality.

The PDLP solver does not generate a basic solution. Currently it has no crossover procedure to obtain one.

When modifications have been made to an LP problem, the simplex solver (only) can solve the modified problem from the optimal basis of the original LP, a process known as "hot starting".

MIP

The HiGHS MIP solver uses established branch-and-cut techniques. It is largely single-threaded, although implementing a multi-threaded tree search is work in progress.

QP

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