integer linear programming modeling

This paper studies the optimization of Integer Linear-Exponential Programming (ILEP): maximizing/minimizing a linear-exponential objective function subject to linear-exponential constraints involving the exponential function (x mapsto 2^x) and the modulo operation ((x,y) mapsto x mod 2^y). As the decision version is NP-complete, we propose—first in the literature—the *Integer Linear-Exponential Straight-Line Program* (ILESLP) as a succinct representation of optimal solutions, placing the problem in an extended NPO class and circumventing standard binary search frameworks. Our method models solution structure via arithmetic circuits and leverages an integer factorization oracle to design polynomial-time algorithms for feasibility verification and objective-value comparison. The core contribution is a proof that every optimal solution admits an ILESLP representation of polynomial size, and that both feasibility and dominance (i.e., objective-value comparison) can be verified in polynomial time. This establishes ILEP as efficiently certifiable within its extended complexity class.

To address the low computational efficiency of solving mixed-integer bilevel linear programs (MIBLPs), this paper proposes a novel unified modeling approach based on *improving directions*: a single subproblem simultaneously verifies bilevel feasibility and generates strong valid inequalities. Theoretically, we characterize the role of improving directions in encoding the follower’s optimality conditions, establish an optimality-based relaxation hierarchy, and extend the theory of continuous cutting-plane closures to the mixed-integer bilevel setting. Algorithmically, we integrate improving-direction analysis into a branch-and-cut framework, implementing it atop the open-source solver MibS. Computational experiments demonstrate that our method substantially enhances inequality strength and overall solution performance across standard benchmark instances.

This work addresses integer and mixed-integer nonlinear programming (INLP/MINLP) problems, aiming to accelerate exact algorithms—particularly branch-and-bound (BB)—while rigorously preserving global optimality. We propose a unified learnable BB framework that jointly integrates supervised learning, imitation learning, and reinforcement learning into four core components: branching variable selection, cutting-plane generation, node prioritization, and parameter tuning. The framework is agnostic to variable types—supporting discrete, continuous, and hybrid structures—and is validated on real-world applications including unit commitment, vehicle routing, and hydroelectric scheduling. Key contributions include: (i) the first taxonomy of learning-augmented optimization methods organized along both solver architecture and learning paradigm dimensions; (ii) substantial convergence acceleration without compromising solution quality or optimality guarantees; and (iii) advancement toward scalable, generalizable intelligent optimization solvers.

This paper investigates the parameterized complexity of Integer Linear Programming (ILP) with lower and upper bounds, parameterized by the “distance to generalized matching”—the minimum number of modifications required to transform the constraint matrix into one where each column has ℓ₁-norm at most two (encompassing polynomial-time solvable matching and flow problems). The authors introduce two novel structural parameters: a *variable backdoor* (minimum column deletions to achieve the generalized matching structure) and a *constraint backdoor* (minimum row deletions). They present the first fixed-parameter tractable (FPT) algorithm parameterized by variable backdoor size (p). In contrast, they prove W[1]-hardness parameterized by constraint backdoor size (h) and devise a randomized XP algorithm for unary-encoded instances. Key technical innovations include a variant of lattice convexity, Graver basis-enhanced local search, and a pseudo-polynomial reduction to Exact Matching—enabling both tight complexity classification and significant algorithmic advances.

To address the low online solving efficiency of Mixed-Integer Programming (MIP), this paper proposes PreMIO: a framework that pretrains lightweight machine learning models on offline data to devise the first data-driven, multi-variable branching strategy with both theoretical provability and interpretability. Leveraging concentration inequalities, the strategy guides hyperplane-based cuts to dynamically partition the feasible region—bridging the long-standing gap between theoretical guarantees and engineering practicality in ML-augmented MIP. PreMIO seamlessly integrates with mainstream MIP solvers without modifying their core algorithms. Evaluated on standard operations research benchmarks (e.g., MIPLIB) and real-world industrial instances, PreMIO reduces average solving time by 32%–57% and significantly decreases the number of explored nodes, demonstrating strong generalization and deployment feasibility.

This work addresses the challenge that large-scale or highly constrained mixed-integer programming (MIP) problems are often intractable for existing commercial solvers. To overcome this limitation, the authors propose integrating the Random-Key Optimizer (RKO) framework into MIP solving. The approach conducts search in a continuous random-key space and employs a problem-specific decoder to map candidate solutions into feasible integer solutions, thereby decoupling the search process from feasibility enforcement. This separation substantially reduces the effective search space and enhances both solution feasibility and convergence speed. Experimental results on portfolio optimization and time-dependent traveling salesman problems demonstrate that RKO consistently outperforms leading commercial solvers in both solution quality and computational efficiency, exhibiting strong scalability and competitive performance.

This paper studies covering-type mixed-integer linear programming (CMILP) problems with a fixed number of constraints, encompassing classical models such as multidimensional knapsack covering, facility location, and supplier selection. Methodologically, we leverage polyhedral vertex structure analysis to decompose the problem into a family of multidimensional knapsack covering subproblems—each involving only one continuous variable—and integrate linear programming relaxation with tailored approximation schemes. We further derive a compact, theoretically optimal linear formulation. Our main contributions are the first polynomial-time approximation scheme (PTAS) and fully polynomial-time approximation scheme (FPTAS) for CMILP under a fixed constraint count, breaking the long-standing 2-approximation barrier for the single-constraint case. Notably, our FPTAS for the single-constraint setting achieves both scalability and provable accuracy guarantees, significantly extending the tractable problem size and solution quality.

Traditional mixed-integer linear programming (MILP) modeling struggles to accommodate the diversity of complex combinatorial optimization problems, resulting in cumbersome formulations and limited solver compatibility. This work proposes OptiDSL, a novel domain-specific language (DSL)-centric modeling paradigm that leverages large language models to automatically translate natural language descriptions into structured DSL representations, thereby decoupling modeling from solving. By moving beyond MILP constraints, the framework enables flexible integration of diverse specialized, heuristic, and learning-based solvers. Evaluated across 44 combinatorial optimization problem classes, OptiDSL achieves a 51.66% improvement in modeling accuracy and reduces modeling time by 91.71% compared to conventional MILP approaches, while further boosting solution accuracy by 23.09% over existing benchmarks.

This work addresses key challenges in integer linear programming (ILP) solvers—limited generalization, reliance on external solvers, and poor efficiency in multimodal energy landscapes—by proposing a training-free, solver-free sampling-based optimization framework that directly explores the discrete feasible region. Leveraging the linear structure of ILP, the method designs a proposal distribution satisfying detailed balance and incorporates a dual tempering mechanism that jointly modulates temperature and penalty parameters to dynamically adjust constraint barriers while preserving the original objective function, thereby enhancing global exploration. Experiments demonstrate that the approach consistently outperforms SCIP across four benchmarks, matches or exceeds Gurobi’s performance on two tasks within 200 seconds, exhibits superior robustness on out-of-distribution instances, and competes effectively with classical solvers on MIPLIB 2017 without any parameter tuning.