Interior-point method

Overview

An interior-point method (IPM) solves a constrained optimisation problem by replacing each inequality with a logarithmic barrier term. The barrier grows without bound as an iterate approaches a constraint, so the iterates stay strictly inside the feasible region. The review writes the barrier problem for a convex QP with one parameter \mu > 0 2510.21773 §2 eq. 3. Equality constraints are kept as equalities and are not put in the barrier 2510.21773 §2 eq. 3.

The invariant the variants share is that the iterates stay strictly interior, so every inequality holds with a positive slack 2510.21773 §2. The variants differ in how they solve the linear systems that arise, and in how much of the problem's sparsity they exploit.

This page rests on a single card: a review of real-time convex QP solvers for legged robots 2510.21773. The review runs no new experiments. Its claims about interior-point solvers come from its own qualitative tables and from benchmarks it relays 2510.21773 §3. Use the review as a map of the family, then check the solver's own documentation and the original benchmarks before relying on a number.

Variants

Structure-exploiting interior-point solver (HPIPM)

This variant is for optimal-control QPs, the sparse, block-structured problems that arise in model predictive control over a long horizon. The review names HPIPM as the structure-exploiting interior-point solver and rates it as warm-startable, early-terminable and suitable for real-time use 2510.21773 tab. 1. It accepts the optimal-control (OCP) form of the problem 2510.21773 tab. 2. The review's guidance is to use it for long-horizon MPC 2510.21773 §4. Card: Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots.

The review says the family handles "block-banded or tree" structure in the dynamics 2510.21773 tab. 1. It does not describe the factorisation HPIPM uses, so this page does not either.

Embedded sparse interior-point solvers (qpSWIFT, PIQP)

These are the other two interior-point solvers the review names for robot control. qpSWIFT is a sparse primal-dual interior-point solver that the review lists for embedded sparse robotics QPs 2510.21773 tab. 1. PIQP is a proximal interior-point solver that the review lists for ill-conditioned sparse QPs and for embedded constraints 2510.21773 tab. 1. Both are rated for real-time use in the review's table, which is a judgement and not a measurement 2510.21773 tab. 1.

PIQP's proximal term is not defined in the review, so this page does not describe how it changes the barrier problem 2510.21773 §2.

Generic primal-dual interior-point solvers

The review lists a generic primal-dual interior-point family. It is rated as not warm-startable and not suited to real-time use, and the review's best scenario for it is "offline or high-accuracy checks; general sparse QPs" 2510.21773 tab. 1. Gurobi and MOSEK appear in the review's interior- point group with sparse matrix support. Both are listed without warm start or hot start, and both support early termination and dual-gap reporting 2510.21773 tab. 2.

Conic interior-point solver (Clarabel)

Clarabel is a conic interior-point solver that also handles QPs as a special case. The review proposes it when conic constraints or optimality certificates are needed 2510.21773 §2 2510.21773 tab. 1. It is not a dedicated QP solver, and the review gives no timing comparison for it against the QP-specific solvers 2510.21773 §3.

When to use which

The review gives no memory, latency-on-target or hardware-cost figures for the interior-point family. The table records only what the review's ratings and cited results support.

The review's benchmark claims for this family are relayed from another study, which the review cites as Stark et al. (arXiv:2502.01329). In that MPC benchmark, as the review summarises it, HPIPM had the lowest solve time on sparse QPs, while OSQP was robust but slower at high accuracy 2510.21773 §3. No IPM-versus-IPM timing in the review is measured by its authors.

Implementation sketch

The review states the barrier problem and nothing about the iteration. It gives no Newton or KKT update for the barrier problem, no barrier-parameter schedule and no stopping test 2510.21773 §2 eq. 3. The sketch below covers only the barrier objective. Solver-specific details must come from each solver's own documentation.

# Barrier objective for a convex QP with inequalities C x < u  [§2 eq. 3]
# Input:  x (n,); H (n, n); q (n,); C (n_i, n); u (n_i,); mu > 0
# Output: barrier objective value, or infinity outside the interior

def barrier_objective(x, H, q, C, u, mu):
    # Slack of each inequality: must stay strictly positive  [§2 eq. 3]
    slack = u - C @ x
    # Outside the strict interior the barrier is undefined  [§2]
    if any(slack <= 0):
        return inf
    # Quadratic and linear part of the QP objective  [§2 eq. 3]
    quad = 0.5 * x @ H @ x + q @ x
    # Logarithmic barrier, weighted by mu  [§2 eq. 3]
    return quad - mu * sum(log(slack))

Contradictions

None known. The review's tables agree on the warm-start and early-termination entries for the interior-point rows they share 2510.21773 tab. 1 2510.21773 tab. 2. The review's ratings are its own judgements, not measurements 2510.21773 tab. 1.

Papers

• Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots: compares convex QP solver families for legged-robot MPC and whole-body control, and rates the interior-point solvers in its tables.