---
kind: concept
slug: interior-point-method
title: Interior-point method
summary: >-
  Solve a convex QP by keeping iterates strictly inside the inequality constraints through a log barrier. For long-horizon sparse MPC, a structure-exploiting interior-point solver (HPIPM) is the review's recommendation. Only one review card supports this page, and its benchmark claims are relayed from other work.
tags: [optimization, quadratic-programming, control, real-time]
papers: ["2510.21773"]
related: [active-set-method, model-predictive-control]
status: draft
updated_at: 2026-10-25
---

## 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 $\mu > 0$ [arxiv:2510.21773 §2 eq. 3]. Equality
constraints are kept as equalities and are not put in the barrier
[arxiv: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 [arxiv: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 [arxiv: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 [arxiv: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
[arxiv:2510.21773 tab. 1]. It accepts the optimal-control (OCP) form of the
problem [arxiv:2510.21773 tab. 2]. The review's guidance is to use it for
long-horizon MPC [arxiv:2510.21773 §4]. Card: [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773).

The review says the family handles "block-banded or tree" structure in the
dynamics [arxiv: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 [arxiv:2510.21773 tab. 1]. PIQP is a
proximal interior-point solver that the review lists for ill-conditioned
sparse QPs and for embedded constraints [arxiv:2510.21773 tab. 1]. Both are
rated for real-time use in the review's table, which is a judgement and not a
measurement [arxiv: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 [arxiv: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"
[arxiv: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
[arxiv: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 [arxiv:2510.21773 §2] [arxiv: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 [arxiv: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.

| Situation | Use | Why |
|---|---|---|
| Long-horizon, sparse MPC or OCP QP; warm start wanted | Structure-exploiting IPM (HPIPM) | Rated warm-startable and real-time, and recommended for long-horizon MPC [arxiv:2510.21773 tab. 1] [arxiv:2510.21773 §4] |
| Embedded sparse robot QP | qpSWIFT | Listed for embedded sparse robotics QPs [arxiv:2510.21773 tab. 1] |
| Ill-conditioned sparse QP, or embedded constraints | PIQP | Listed for these cases [arxiv:2510.21773 tab. 1] |
| Conic constraints or certificates needed | Clarabel | Listed for these cases [arxiv:2510.21773 tab. 1] |
| General sparse QP, offline or as a high-accuracy check | Generic primal-dual IPM, e.g. Gurobi or MOSEK | Rated not real-time and not warm-startable [arxiv:2510.21773 tab. 1] [arxiv:2510.21773 tab. 2] |
| Small or medium dense QP, such as whole-body control | Not an interior-point solver here | The review rates dense active-set solvers and ProxQP for this shape [arxiv:2510.21773 §3] [arxiv:2510.21773 §4]. See [Active-set method](/concepts/active-set-method) |

The review's benchmark claims for this family are relayed from another study,
which the review cites as Stark et al. ([arXiv:2502.01329](https://arxiv.org/abs/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
[arxiv: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 [arxiv:2510.21773 §2 eq. 3]. The sketch below covers only
the barrier objective. Solver-specific details must come from each solver's
own documentation.

```pseudocode
# 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 [arxiv:2510.21773 tab. 1]
[arxiv:2510.21773 tab. 2]. The review's ratings are its own judgements, not
measurements [arxiv:2510.21773 tab. 1].

## Papers

- [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773): compares convex QP solver families for legged-robot MPC and whole-body control, and rates the interior-point solvers in its tables.