Quadratic programming

Overview

A quadratic program (QP) asks for the decision vector that minimises a quadratic function subject to linear constraints. The review writes the convex case as minimising ½xᵀHx + qᵀx subject to Ax = b and Cx ≤ u, with H positive semi-definite 2510.21773 §2. Equality rows are A and b; inequality rows are C and u.

QPs appear inside larger controllers. Model predictive control (MPC) and whole-body control (WBC) on legged robots both reduce to QPs that must be solved within tight time and energy budgets. MPC gives sparse problems with optimal-control structure over long horizons. WBC gives small-to-medium dense problems at 100 to 1000 Hz 2510.21773 §1. A walking controller that solves one such QP per control step reports an average control rate of 1 kHz on a simulated humanoid 1311.1839 §1 1311.1839 §5. That is author-reported, from one model.

Other robotics cards use a QP as one part of a larger method:

• Inverse kinematics for teleoperation. A QP-based solve picks joint velocities from weighted task errors, a regulariser and a collision term 2411.07534 §II eq. 3. The printed task term is a norm, and a QP needs the squared form, so check which one is meant before implementing.
• Whole-body control on a humanoid. A momentum-based controller, framed as a QP, outputs joint accelerations and contact wrenches [arxiv:2411.01014 §2.8]. The paper gives no gains, task weights or rates.
• Contact forces on a quadruped. A regularised least-squares problem with force bounds and a friction cone gives the stance forces 2412.09816 §3 eq. 12 eq. 13 eq. 14. The paper solves it with its own method, not a generic QP solver (see Variants).
• Trajectory smoothing. After a search planner, a quadratic cost on the squared n-th derivative is minimised under equality constraints at the endpoints and waypoints 1709.05401 §4 eq. 13.

When the problem is nonlinear, sequential quadratic programming (SQP) solves a QP built around the current iterate at each step 2510.21773 §2. The review does not say how the Hessian approximation in that QP is formed.

The first five subsections under Variants follow the review's solver taxonomy. The last two cover a direct solve for equality-only problems, and a cone constraint handled by projection, which is not a solver family in the review.

Variants

Active-set methods

These guess which inequalities hold with equality at the solution, and solve an equality-constrained system on that guess. They warm-start well when the guessed set changes slowly, but their timing can jitter when the set switches 2510.21773 §2 2510.21773 tab. 1. Named solvers include qpOASES, Eiquadprog, qpmad and DAQP 2510.21773 §2. See Active-set method.

In a humanoid walking controller, one QP per control step is solved by a custom active-set solver that starts from the previous step's active set. The paper reports one linear solve in about 97% of steps 1311.1839 §1 fig. 2, and an average QP solve of 0.2 ms, against 1.0 ms for Gurobi's dual simplex and 2.2 ms for CVXGEN on the same walk 1311.1839 tab. 1. These are author-reported, from one simulated 34-DOF model on a laptop [arxiv:1311.1839 §5.1]. QP setup took about 0.8 ms of a 1 ms control step, so the solve is not the main cost 1311.1839 §5. The paper cannot guarantee finite termination. It falls back to an interior-point solver that it does not name 1311.1839 §4.

Card: Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots. Card: An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion.

Interior-point methods

These replace each inequality with a logarithmic barrier weighted by a parameter μ > 0, so iterates stay strictly inside the constraints 2510.21773 §2 eq. 3. Named solvers include HPIPM, which exploits block structure, along with qpSWIFT and PIQP 2510.21773 §2. See Interior-point method. Card: Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots.

Operator splitting (ADMM)

Each inequality gets a slack variable s ≥ 0, so that Cx + s = u 2510.21773 §2 eq. 4. The review gives this reformulation only. It does not give the ADMM iterations, the penalty update or a termination test 2510.21773 §2. OSQP is the representative solver 2510.21773 §2. Card: Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots.

Augmented Lagrangian and proximal methods

These use a penalised Lagrangian with multipliers for equalities and inequalities, and penalty weights ρ_e and ρ_i > 0. The review gives the multiplier updates for both, with the inequality multiplier kept non-negative 2510.21773 §2 eq. 5 2510.21773 §2 eq. 6 2510.21773 §2 eq. 7. ProxQP is the representative solver 2510.21773 §2. Card: Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots.

Code generation

For a problem whose structure is fixed and only its numbers change, a code generator emits specialised C code. CVXGEN, CVXPYgen and FORCESPRO are the named tools 2510.21773 §2. The review rates CVXGEN and CVXPYgen for tiny problems whose layout never changes 2510.21773 tab. 1.

Conic solver

When conic constraints or optimality certificates are needed, the review proposes Clarabel, a conic solver that includes QPs as a special case 2510.21773 §2. See Interior-point method for where it sits in the family.

Equality-only problems (direct solve)

With only equality constraints, the QP reduces to one linear KKT system for the decision and the multipliers 2510.21773 §2 eq. 1. The review rates this direct solve as real-time, and names equality-only QPs with fixed structure as its case 2510.21773 tab. 1. A planner's refinement step is of this kind: it minimises the integral of the squared n-th derivative, with equality constraints at the endpoints and waypoints 1709.05401 §4 eq. 13.

Cone constraints by projection (not a generic QP)

A generic QP solver needs a polyhedral stand-in for a cone. One quadruped controller keeps the exact friction cone and solves its contact-force problem with a custom projected-gradient method, GPGD 2412.09816 §4. Each pass takes a Newton-type step with the Hessian 2Q, then projects each stance foot's force onto the normal-force bounds and the friction cone [arxiv:2412.09816 alg. 1]. Against qpOASES, the paper reports GPGD 2.47 times faster, with lower residual and constraint violation 2412.09816 tab. 1 §6.1.2. The comparison is not like for like. The GPGD implementation is naive, the qpOASES one is optimised, and the two use different constraint sets, a cone and a linear pyramid 2412.09816 §6.1.2. As printed, alg. 1 never reassigns its iterate, so an implementation must add that update 2412.09816 alg. 1. The other paper keeps a polyhedral friction cone inside a QP 1311.1839 §3 eq. 9.

When to use which

The review's ratings are its own qualitative judgements, not measurements. Its tables label real-time suitability as scenario-dependent 2510.21773 tab. 1. The table below records what its guidance supports, and what the other cards report.

Many of the review's timing claims are relayed from one benchmark it cites for both MPC and WBC, plus a dense-QP benchmark 2510.21773 §3. The review measures none of them itself. Check the original benchmark before choosing on a timing number.

Implementation sketch

The review does not give a general QP solver loop. The sketch below encodes its solver-selection advice by problem shape. It is the review's advice, not an algorithm the review states 2510.21773 §3 2510.21773 §4.

# Candidate solver families by problem shape  [arxiv:2510.21773 §3] [§4]
# Input:  shape, one of 'ocp', 'dense_wbc', 'large_sparse', 'fixed',
#         'conic', 'ill_conditioned'
# Output: candidate solver names (suggestions, not a ranking)

def candidate_solvers(shape):
    # Sparse optimal-control structure over a long horizon  [§4]
    if shape == 'ocp':
        return ['HPIPM']
    # Small or medium dense problems at high rate  [§3] [§4]
    if shape == 'dense_wbc':
        return ['qpmad', 'DAQP', 'qpOASES', 'ProxQP']
    # Large sparse problems, repeated solves, moderate accuracy  [§3]
    if shape == 'large_sparse':
        return ['OSQP']
    # Fixed structure, tiny, generated C code  [tab. 1]
    if shape == 'fixed':
        return ['CVXGEN', 'CVXPYgen', 'FORCESPRO']
    # Conic constraints or certificates  [§2]
    if shape == 'conic':
        return ['Clarabel']
    # Ill-conditioned sparse problems  [tab. 1]
    if shape == 'ill_conditioned':
        return ['PIQP']
    return []

Notes on the sketch. The review's own tables disagree on some solver properties (see Contradictions), so the list is a starting set to test, not a ranking. Solver timings are not in the sketch because the review gives them only for specific hardware and problem sizes 2510.21773 §3. The sketch covers only the review's shapes. It has no entry for a cone handled by projection 2412.09816 §4, because that method is not a solver in the review.

Contradictions

• Gurobi and warm start. The walking-controller paper reports its Gurobi baselines as warm-started, with both the barrier and dual simplex methods 1311.1839 §5.1. The review's table marks Gurobi, an interior-point solver, as not warm-startable 2510.21773 tab. 2. This page does not resolve it. Check the solver version and settings before comparing timings.
• qpOASES matrix support. The review's first table describes qpOASES as small-to-medium dense 2510.21773 tab. 1. Its second table lists it as dense or sparse 2510.21773 tab. 2. The review does not reconcile the two, so check the solver's own documentation before choosing it for sparse input.

Papers

• Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots: a review of convex QP solver families for legged-robot MPC and whole-body control, with a taxonomy and selection guidance.
• An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion: a humanoid walking controller that solves a convex QP each control step with a custom warm-started active-set solver.
• Quadrotor lattice planning with LQMT costs: a quadrotor planner whose search is followed by a QP that smooths the trajectory through waypoints.
• Effective Virtual Reality Teleoperation of an Upper-body Humanoid with Modified Task Jacobians and Relaxed Barrier Functions for Self-Collision Avoidance: a teleoperation IK solved as a QP, with a soft relaxed-barrier term for self-collision.
• Mixed Reality Teleoperation Assistance for Direct Control of Humanoids: a teleoperation system whose whole-body controller is framed as a QP; the paper gives no QP settings.
• Distributed Inverse Dynamics Control for Quadruped Robots using Geometric Optimization: contact forces from a cone-constrained problem solved by a custom projected-gradient method, compared with qpOASES.