We propose FlexQP, an always-feasible convex quadratic programming (QP) solver based on an $\ell_1$ elastic relaxation of the QP constraints. If the original constraints are feasible, FlexQP provably recovers the optimal solution. If the constraints are infeasible, FlexQP identifies a solution that minimizes the constraint violation while keeping the number of violated constraints sparse. Such infeasibilities arise naturally in sequential quadratic programming (SQP) subproblems due to the linearization of the constraints. We prove the convergence of FlexQP under mild coercivity assumptions, making it robust to both feasible and infeasible QPs. We then apply deep unfolding to learn LSTM-based, dimension-agnostic feedback policies for the algorithm parameters, yielding an accelerated Deep FlexQP. To preserve the exactness guarantees of the relaxation, we propose a normalized training loss that incorporates the Lagrange multipliers. We additionally design a log-scaled loss for PAC-Bayes generalization bounds that yields substantially tighter performance certificates, which we use to construct an accelerated SQP solver with guaranteed QP subproblem performance. Deep FlexQP outperforms state-of-the-art learned QP solvers on a suite of benchmarks including portfolio optimization, classification, and regression problems, and scales to dense QPs with over 10k variables and constraints via fine-tuning. When deployed within SQP, our approach solves nonlinear trajectory optimization problems 4-16x faster than SQP with OSQP while substantially improving success rates. On predictive safety filter problems, Deep FlexQP reduces safety violations by over 70\% and increases task completion by 43\% compared to existing methods.
We propose FlexQP, an always-feasible convex quadratic programming (QP) solver based on an ℓ1 elastic relaxation of the QP constraints. If the original constraints are feasible, FlexQP provably recovers the optimal solution. If the constraints are infeasible, FlexQP identifies a solution that minimizes the constraint violation while keeping the number of violated constraints sparse. Such infeasibilities arise naturally in sequential quadratic programming (SQP) subproblems due to the linearization of the constraints. We prove the convergence of FlexQP under mild coercivity assumptions, making it robust to both feasible and infeasible QPs. We then apply deep unfolding to learn LSTM-based, dimension-agnostic feedback policies for the algorithm parameters, yielding an accelerated Deep FlexQP. To preserve the exactness guarantees of the relaxation, we propose a normalized training loss that incorporates the Lagrange multipliers. We additionally design a log-scaled loss for PAC-Bayes generalization bounds that yields substantially tighter performance certificates, which we use to construct an accelerated SQP solver with guaranteed QP subproblem performance. Deep FlexQP outperforms state-of-the-art learned QP solvers on a suite of benchmarks including portfolio optimization, classification, and regression problems, and scales to dense QPs with over 10k variables and constraints via fine-tuning. When deployed within SQP, our approach solves nonlinear trajectory optimization problems 4-16x faster than SQP with OSQP while substantially improving success rates. On predictive safety filter problems, Deep FlexQP reduces safety violations by over 70% and increases task completion by 43% compared to existing methods.
Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.
At the intersection of combinatorial and nonlinear optimization, quadratic programming (QP) plays an important role in practice and optimization theory.With wide-spread real-world applications including, but not limited to, engineering design, game theory, and economics, quadratic programming has many valuable contributions and has attracted considerable interest over the past few decades.As a more general formulation, quadratically constrained quadratic programming (QCQP) problems arise naturally from a lot of applications such as facility location, production planning, economic equilibria and Euclidean distance geometry, etc.Most global optimization approaches for nonconvex QCQPs are based on constructing a convex or linear relaxation of the problem, and embedding it within an exhaustive search mechanism, where a lower bound is calculated as the value of the objective function of this relaxation (for a minimization problem) at each iteration of the search tree.The two principle elements affecting the overall effectiveness of such exhaustive search methods are the tightness of the underlying relaxation and the efficiency of calculating the lower bound.Besides, the performance of a branch-and-bound algorithm also relies on the procedure used to select a branching node and branching variable, which leads to good quality feasible solutions or equivalently an upper bound.vii LIST OF TABLES 5.5 Computational results of 0-1 QPs with n = 40 . . . . . . . . . . . .110 5.6 Computational results of 0-1 QPs with n = 50 . . . . . . . . . . . .
We present a novel branch-and-bound solver for mixed-integer quadratic programs (MIQPs) that efficiently exploits the first-order OSQP solver for the quadratic program (QP) sub-problems. Our algorithm is very robust, requires no dynamic memory allocation and is division-free once an initial factorization is computed. Thus, it suitable for embedded applications with low computing power. Moreover, it does not require any assumption on the problem data such as strict convexity of the objective function. We exploit factorization caching and warm-starting to reduce the computational cost of QP relaxations during branch-and-bound and over repeated solutions of parametric MIQPs such as those arising in embedded control, portfolio optimization, and machine learning. Numerical examples show that our method, using a simple high-level Python implementation interfaced with the OSQP solver, is competitive with established commercial solvers.
We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP’s dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP .
Machine Learning (ML) optimization frameworks have gained attention for their ability to accelerate the optimization of large-scale Quadratically Constrained Quadratic Programs (QCQPs) by learning shared problem structures. However, existing ML frameworks often rely heavily on strong problem assumptions and large-scale solvers. This paper introduces NeuralQP, a general hypergraph-based framework for large-scale QCQPs. NeuralQP features two main components: Hypergraph-based Neural Prediction, which generates embeddings and predicted solutions for QCQPs without problem assumptions, and Parallel Neighborhood Optimization, which employs a McCormick relaxation-based repair strategy to identify and correct illegal variables, iteratively improving the solution with a small-scale solver. We further prove that our framework UniEGNN with our hypergraph representation is equivalent to the Interior-Point Method (IPM) for quadratic programming. Experiments on two benchmark problems and large-scale real-world instances from QPLIB demonstrate that NeuralQP outperforms state-of-the-art solvers (e.g., Gurobi and SCIP) in both solution quality and time efficiency, further validating the efficiency of ML optimization frameworks for QCQPs.
Low-thrust trajectory optimization is a central task in interplanetary mission design, but its nonlinear dynamics and operational constraints often lead to challenging non-convex optimal-control problems. Sequential convex programming has emerged as an effective approach to address these problems, while quantum annealing offers a complementary paradigm for solving quadratic unconstrained binary optimization problems. This paper introduces quSCP, a quantum-based sequential convex programming framework that reformulates each convex subproblem as a quadratic unconstrained binary optimization problem suitable for quantum and hybrid quantum--classical solvers. Equality, inequality, and trust-region constraints are embedded through quadratic penalty terms, while an iterative refinement strategy is used to reduce the accuracy loss introduced by binary discretization. The method is assessed on a fuel-optimal Earth--Mars low-thrust transfer by comparing standard sequential convex programming, a continuous quadratic unconstrained formulation, direct quantum processing unit sampling, and D-Wave hybrid solvers. Results show that quSCP produces physically consistent trajectories with propellant consumption and nonlinear constraint violations close to classical benchmarks. Direct quantum annealing is feasible only for small instances because of embedding overhead and hardware connectivity limits, whereas hybrid solvers scale to larger discretizations. Although no computational quantum advantage is demonstrated with current hardware, the results show that quantum and hybrid quantum--classical optimization can already provide competitive solutions for demanding trajectory design problems.