A convex QP solver learns its own tuning: under 8 hours on a laptop, fastest on about 70% of Maros–Mészáros instances

A regularized interior-point solver for convex quadratic programs can now have its hyperparameters adjusted during the run by a learned policy instead of a fixed hand-tuned contraction rule. Previously the regularization weights, barrier parameter and inner tolerance of such solvers were set by hand or by heuristics; theory only requires the weights to stay positive and bounded, the barrier parameter to vanish and the inner tolerance to be summable, without saying which values to use. On the Maros–Mészáros test set, RL-QProxIP is the fastest solver on about 70% of instances, and QProxIP needs about 3.5 times the time to reach the same fraction; on final coverage the source says in one place that QProxIP solves 100% of instances while RL-QProxIP saturates near 92%, and in another about 96% and about 85%, so the two accounts disagree and no coverage ranking can be drawn.

The policy reads a state built from normalized primal and dual residuals, the barrier parameter and the inner tolerance, and outputs five contraction factors between zero and one. The actions act through a saturated linear decrease on the regularization weights, barrier parameter and inner tolerance, following the update form used by solvers such as QPALM, ProxQP and PIQP, while the underlying interior-point algorithm is left unchanged. Training used 800k randomly generated small quadratic programs with 5 to 50 variables, including duplicated and nearly duplicated constraints and controlled effective condition numbers, and finished in under 8 hours on a laptop; the validation set holds 100 same-distribution problems with 5 to 47 variables and median 17.

The learned policy still speeds up solves outside its training distribution, and its inference cost is small relative to solve time. Training covered only small random problems, so performance on large benchmarks is zero-shot generalization rather than a directly optimized target. On the 100 smallest Maros–Mészáros instances, RL-QProxIP solves half of them within about 44 ms, roughly 2.3 times faster than QProxIP; on the 73-instance subset that at least one of the two solvers finishes within one second, it is the fastest on 74% of problems against 26% for QProxIP, and neural-policy inference costs about 1 ms per problem.

The demonstrated scope is the authors' own regularized primal-dual interior-point solver QProxIP and its reinforcement-learning extension RL-QProxIP, trained only on small random quadratic programs with 5 to 50 variables and evaluated on a same-distribution validation set, a 10x scaled random set, and the 100 smallest Maros–Mészáros instances selected by folder size. The authors propose future directions including large-scale sparse problems from power systems, optimal control problems with a three-banded structure, financial optimization with volatile market data, and transferring the policy to other second-order schemes such as primal interior-point methods.

Training was restricted to small random quadratic programs, validation stayed within a single interior-point solver scheme, and the trained policy's entropy plateaus at a relatively high level, indicating it has not converged to near-deterministic behavior. The authors also note that on problems solved in only a few iterations the learned policy may add limited benefit and its inference overhead can partly offset faster convergence. The source gives two conflicting accounts of final coverage on the full Maros–Mészáros set, so which solver covers more instances remains undetermined.