Zedong Peng

This paper presents MPAX (Mathematical Programming in JAX), a versatile and efficient toolbox for integrating linear programming (LP) into machine learning workflows. MPAX implemented the state-of-the-art first-order methods, restarted average primal-dual hybrid gradient and reflected restarted Halpern primal-dual hybrid gradient, to solve LPs in JAX. This provides native support for hardware accelerations along with features like batch solving, auto-differentiation, and device parallelism. Extensive numerical experiments demonstrate the advantages of MPAX over existing solvers. The solver is available at https://github.com/MIT-Lu-Lab/MPAX.

We introduce cuPDLPx, a further enhanced GPU-based first-order solver for linear programming. Building on the recently developed restarted Halpern PDHG for LP, cuPDLPx incorporates a number of new techniques, including a new restart criterion and a PID-controlled primal weight update. These improvements are carefully tailored for GPU architectures and deliver substantial computational gains. Across benchmark datasets, cuPDLPx achieves 2.5x-5x speedups on MIPLIB LP relaxations and 3x-6.8x on Mittelmann’s benchmark set, with particularly strong improvements in high-accuracy and presolve-enabled settings. The solver is publicly available at https://github.com/MIT-Lu-Lab/cuPDLPx.

Quantum algorithms have shown promise in solving Quadratic Unconstrained Binary Optimization (QUBO) problems, benefiting from their connection to the transverse field Ising model. Various Ising solvers, both classical and quantum, have emerged to tackle such problems efficiently but lack global optimality guarantees and often suffer from hardware limitations such as limited qubit availability. In this work, we propose a hybrid branch-and-bound (B&B) framework that integrates Ising solvers as heuristics within a classical B&B algorithm. Unlike prior theoretical studies, our work presents a practical implementation, available as open-source on GitHub. We explore when and where to apply Ising solvers in the search tree and introduce a custom branching rule optimized QUBO embedding. Our method is evaluated on hundreds of QUBO instances from QUBOLib.jl using Gurobi and the D-Wave quantum annealer. Our results show up to 11% less solution time and 17% fewer nodes compared to default Gurobi, an off-the-shelf commercial optimization solver. These findings demonstrate the value of hybrid quantum-classical strategies for enhancing exact optimization.

Advancements in convexification and domain reduction techniques have significantly enhanced the performance of global optimization solvers for challenging Mixed-Integer Nonlinear Programming (MINLP) problems. However, the great potential of these techniques has not yet been fully harnessed in decomposition-based MINLP solvers. Motivated by this consideration, we present an improved Outer-Approximation (OA) method by integrating convexification and bound tightening techniques. To effectively solve convex MINLP problems, both multi-tree and single-tree (i.e., known as the LP/NLP Branch-and-Bound (B&B) method) settings of the OA method are investigated. Moreover, the advanced global OA and global LP/NLP B&B methods are extended to include these techniques for non-convex MINLP problems. These new methods have been developed and incorporated into the open-source Mixed-Integer Nonlinear Decomposition Toolbox for Pyomo-MindtPy. Extensive computational experiments were conducted using the MINLP benchmark library MINLPLib to validate the effectiveness and reliability of the developed methods. The results show that incorporating the convexification and bound tightening techniques into the (global) OA and (global) LP/NLP B&B methods significantly reduces the computational time and the number of iterations required for the solution.

In this work, we extend the regularization framework from Kronqvist et al. (Math Program 180(1):285–310, 2020) by incorporating several new regularization functions and develop a regularized single-tree search method for solving convex mixed-integer nonlinear programming (MINLP) problems. We propose a set of regularization functions based on distance metrics and Lagrangean approximations, used in the projection problem for finding new integer combinations to be used within the Outer-Approximation (OA) method. The new approach, called Regularized Outer-Approximation (ROA), has been implemented as part of the open-source Mixed-integer nonlinear decomposition toolbox for Pyomo—MindtPy. We compare the OA method with seven regularization function alternatives for ROA. Moreover, we extend the LP/NLP Branch and Bound method proposed by Quesada and Grossmann (Comput Chem Eng 16(10–11):937–947, 1992) to include regularization in an algorithm denoted RLP/NLP. We provide convergence guarantees for both ROA and RLP/NLP. Finally, we perform an extensive computational experiment considering all convex MINLP problems in the benchmark library MINLPLib. The computational results show clear advantages of using regularization combined with the OA method.

Achieving high-quality solutions faster than classical solvers on computationally hard problems is a challenge for quantum optimization to deliver utility. Using a superconducting quantum computer, we experimentally investigate the performance of a hybrid quantum-classical algorithm inspired by semidefinite programming approaches for solving the maximum-cut problem on 3-regular graphs up to several thousand variables. We leverage the structure of the input problems to address sizes beyond what current quantum machines can naively handle. We attain an average approximation ratio of 99% over a random ensemble of thousands of problem instances. We benchmark the quantum solver against similarly high-performing classical heuristics, including the gurobi optimizer, simulated annealing, and the Burer-Monteiro algorithm. A run-time analysis shows that the quantum solver on large-scale problems is competitive against gurobi but short of others on a projected 100-qubit quantum computer. We explore multiple leads to close the gap and discuss prospects for a practical quantum speedup.

Integer semidefinite programming (ISDP) has recently gained attention due to its connection to binary quadratically constrained quadratic programs (BQCQPs), which can be exactly reformulated as binary semidefinite programs (BSDPs). However, it remains unclear whether this reformulation effectively uses existing ISDP solvers to address BQCQPs. To the best of our knowledge, no specialized ISDP algorithms exploit the unique structure of BSDPs derived from BQCQPs. This paper proposes a novel spectral outer approximation algorithm tailored for BSDPs derived from BQCQP reformulations. Our approach is inspired by polyhedral and second-order representable regions that outer approximate the feasible set of a semidefinite program relying on a spectral decomposition of a matrix that simultaneously diagonalizes the objective matrix and an aggregation of the constraint matrices. Computational experiments show that our algorithm is competitive with, and in some cases outperforms, state-of-the-art ISDP solvers such as SCIP-SDP and PAJARITO, highlighting ISDP’s potential for solving BQCQPs.

In shale gas field development, the major endogenous uncertainty comes from the production profile of candidate wells and its realization is dependent on the development decision. In this work, we apply multistage stochastic programming to address the shale gas field development planning problem under production profile uncertainty. Business realities such as budget limit and uncertainty resolution delay are also considered in the proposed model. To solve the multistage stochastic model, a Lagrangean decomposition method and a heuristic method are proposed. Computational results demonstrate the efficiency of the proposed methods on examples that are otherwise intractable. The optimal development strategy from the proposed model is analyzed under different levels of uncertainties. Different cases are further tested to figure out the impact of budget and resolution delay on development decisions.

Long‐term design and planning of shale gas field development is challenging due to the complex development operations and a wide range of candidate locations. In this work, we focus on the multi‐period shale gas field development problem, where the shale gas field has multiple formations and each well can be developed from one of several alternative pads. The decisions in this problem involve the design of the shale gas network and the planning of development operations. A mixed‐integer linear programming (MILP) model is proposed to address this problem. Since the proposed model is a large‐scale MILP, we propose a solution pool‐based bilevel decomposition algorithm to solve it. Results on realistic instances demonstrate the value of the proposed model and the effectiveness of the proposed algorithm.