--- kind: concept slug: quadratic-programming title: Quadratic programming summary: >- Minimise a quadratic objective under linear equality and inequality constraints. Legged-robot MPC and whole-body control reduce to QPs solved at high rates, and QPs also appear in teleoperation IK and trajectory smoothing. Solver families trade warm starts, sparsity, accuracy and code generation. A review maps them to problem shape; its ratings are its own judgements. tags: [optimization, quadratic-programming, control, real-time] papers: ["2510.21773", "1311.1839", "1709.05401", "2411.07534", "2411.01014", "2412.09816"] related: [active-set-method, interior-point-method, model-predictive-control, linear-quadratic-regulator, inverse-kinematics, inverse-dynamics-control] status: draft updated_at: 2026-10-25 --- ## 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 [arxiv: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 [arxiv: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 [arxiv:1311.1839 §1] [arxiv: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 [arxiv: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 [arxiv: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 [arxiv: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 [arxiv: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 [arxiv:2510.21773 §2] [arxiv:2510.21773 tab. 1]. Named solvers include qpOASES, Eiquadprog, qpmad and DAQP [arxiv:2510.21773 §2]. See [Active-set method](/concepts/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 [arxiv: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 [arxiv: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 [arxiv:1311.1839 §5]. The paper cannot guarantee finite termination. It falls back to an interior-point solver that it does not name [arxiv:1311.1839 §4]. Card: [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773). Card: [An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion](/papers/1311.1839). ### Interior-point methods These replace each inequality with a logarithmic barrier weighted by a parameter μ > 0, so iterates stay strictly inside the constraints [arxiv:2510.21773 §2 eq. 3]. Named solvers include HPIPM, which exploits block structure, along with qpSWIFT and PIQP [arxiv:2510.21773 §2]. See [Interior-point method](/concepts/interior-point-method). Card: [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773). ### Operator splitting (ADMM) Each inequality gets a slack variable s ≥ 0, so that Cx + s = u [arxiv: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 [arxiv:2510.21773 §2]. OSQP is the representative solver [arxiv:2510.21773 §2]. Card: [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773). ### 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 [arxiv:2510.21773 §2 eq. 5] [arxiv:2510.21773 §2 eq. 6] [arxiv:2510.21773 §2 eq. 7]. ProxQP is the representative solver [arxiv:2510.21773 §2]. Card: [Real-Time QP Solvers: A Concise Review and Practical Guide Towards Legged Robots](/papers/2510.21773). ### 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 [arxiv:2510.21773 §2]. The review rates CVXGEN and CVXPYgen for tiny problems whose layout never changes [arxiv: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 [arxiv:2510.21773 §2]. See [Interior-point method](/concepts/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 [arxiv: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 [arxiv: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 [arxiv: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 [arxiv: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 [arxiv: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 [arxiv:2412.09816 §6.1.2]. As printed, alg. 1 never reassigns its iterate, so an implementation must add that update [arxiv:2412.09816 alg. 1]. The other paper keeps a polyhedral friction cone inside a QP [arxiv: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 [arxiv:2510.21773 tab. 1]. The table below records what its guidance supports, and what the other cards report. | Situation | Use | Why | |---|---|---| | Small or medium dense QP at high rate, e.g. whole-body control | Dense active-set solver (qpmad, DAQP, qpOASES) or ProxQP | The review's advice for dense WBC. Active-set solvers are rated warm-start capable [arxiv:2510.21773 §4] [arxiv:2510.21773 tab. 1] | | One QP per tick on a humanoid, active set changes little between ticks | Active-set solver warm-started from the previous active set | Author-reported 0.2 ms average solve on one simulated 34-DOF model, one linear solve in about 97% of steps [arxiv:1311.1839 tab. 1] [fig. 2] | | Long-horizon, sparse optimal-control QP (MPC) | Structure-exploiting interior-point solver (HPIPM) | Recommended for long-horizon MPC, and rated warm-startable and real-time [arxiv:2510.21773 §4] [arxiv:2510.21773 tab. 1] | | Large sparse QP, solved repeatedly, moderate accuracy acceptable | Operator splitting (OSQP) | Rated for repeated solves with warm start, but not for high accuracy [arxiv:2510.21773 tab. 1]. Relayed benchmarks say OSQP is robust but slower at high accuracy [arxiv:2510.21773 §3] | | Ill-conditioned sparse QP, or embedded constraints | PIQP | Rated for these cases [arxiv:2510.21773 tab. 1]. ProxQP is rated for small-to-medium WBC and inverse dynamics, not these cases [arxiv:2510.21773 tab. 1] | | Fixed structure, tiny, hard real-time embedded | Code generator (CVXGEN, CVXPYgen, FORCESPRO) | Emits specialised C code for fixed-structure problems [arxiv:2510.21773 §2] [arxiv:2510.21773 tab. 1] | | Conic constraints or optimality certificates | Clarabel | Named for this case [arxiv:2510.21773 §2] [arxiv:2510.21773 tab. 1] | | Offline checks or high-accuracy reference | Generic primal-dual interior-point (Gurobi, MOSEK) | Rated not real-time and not warm-startable [arxiv:2510.21773 tab. 1] | | Exact friction cone at high rate, e.g. quadruped contact forces | Custom projection (GPGD), or a polyhedral cone inside a QP | GPGD is author-reported 2.47 times faster than qpOASES on a pyramid, a comparison the paper says is not like for like [arxiv:2412.09816 §6.1.2]. The other paper uses a polyhedral cone within a QP [arxiv:1311.1839 §3] | | Smoothing a planned trajectory through waypoints | Equality-only QP (direct KKT solve) | The form of [arxiv:1709.05401 §4] [eq. 13]. The paper says the refined trajectory might be unsafe or infeasible, and describes no check [arxiv:1709.05401 §4] | | Teleoperation IK that must return a best-effort command near self-collision | QP-based IK with a relaxed-barrier term in the cost | The paper says the soft term returns best-effort solutions instead of failing [arxiv:2411.07534 §II]. Barrier form and weights are not given, and there is no code [arxiv:2411.07534 §II] [eq. 4] | | Battery-powered robot | Measure energy per solve | The review argues speed alone misses energy use, and names a metric (SFPW). It gives no formula for it [arxiv:2510.21773 §1] [arxiv:2510.21773 §3] | Many of the review's timing claims are relayed from one benchmark it cites for both MPC and WBC, plus a dense-QP benchmark [arxiv: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 [arxiv:2510.21773 §3] [arxiv:2510.21773 §4]. ```pseudocode # 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 [arxiv:2510.21773 §3]. The sketch covers only the review's shapes. It has no entry for a cone handled by projection [arxiv: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 [arxiv:1311.1839 §5.1]. The review's table marks Gurobi, an interior-point solver, as not warm-startable [arxiv: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 [arxiv:2510.21773 tab. 1]. Its second table lists it as dense or sparse [arxiv: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](/papers/2510.21773): 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](/papers/1311.1839): 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](/papers/1709.05401): 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](/papers/2411.07534): 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](/papers/2411.01014): 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](/papers/2412.09816): contact forces from a cone-constrained problem solved by a custom projected-gradient method, compared with qpOASES.