arXiv subjects

Carl Laird

Publications and source records attributed to Carl Laird.

3 recordsLinked to original sources

arXiv2026

Optimization models and algorithms for the Unit Commitment problem

The unit commitment problem determines the optimal strategy to meet the electricity demand at minimum cost by committing power generation units at each point of time. Solving the unit commitment problem gives rise to a challenging optimization problem due to its combinatorial complexity and potentially long solution time requirements. Our proposed solution approach utilizes a decomposition method in conjunction with alternative models from the EGRET library. Results of this decomposition approach tested against four benchmarking systems show that significant computational speed ups are achieved.

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arXiv2026

Efficient Convexification of Kolmogorov-Arnold Networks with Polynomial Functional Forms Via a Continuous Graham Scan Approach

Deterministic global optimization of nonlinear models is important in many scientific and engineering applications. This framework typically involves repeatedly solving convex relaxations of the nonconvex problem, meaning that the strength of the relaxations and the cost of computing them directly determine overall efficiency and solution quality. In this work, we develop a tailored continuous convexification framework for Kolmogorov-Arnold Networks in which the univariate components are polynomial functions. By exploiting the additive separable structure of this architecture, the relaxation problem reduces to computing tight convex envelopes of univariate polynomials. We propose a continuous variant of the classical Graham Scan that constructs these envelopes exactly by identifying the bitangents of the polynomial convex hull without discretization or factorable reformulations. We establish the correctness of the algorithm and characterize its computational complexity, and show how these envelopes can be combined to construct strong convex relaxations for polynomial KANs. Computational results demonstrate that the proposed relaxations are both strong and robust, often producing bounds that are comparable, or even orders of magnitude tighter than relaxations of state-of-the-art global optimization solvers while remaining computationally efficient.

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arXiv2019

Exploring Benefits of Linear Solver Parallelism on Modern Nonlinear Optimization Applications

The advent of efficient interior point optimization methods has enabled the tractable solution of large-scale linear and nonlinear programming (NLP) problems. A prominent example of such a method is seen in Ipopt, a widely-used, open-source nonlinear optimization solver. Algorithmically, Ipopt depends on the use of a sparse symmetric indefinite linear system solver, which is heavily employed within the optimization of barrier subproblems. As such, the performance and reliability of Ipopt is dependent on the properties of the selected linear solver. Inspired by a trend in mathematical programming toward solving larger and more challenging NLPs, this work explores two core questions: first, how does the scalability of available linear solvers, many of which exhibit shared-memory parallelism, impact Ipopt performance; and second, does the best linear solver vary across NLP problem classes, including nonlinear network problems and problems constrained by partial differential equations? To better understand these properties, this paper first describes available open- and closed-source, serial and parallel linear solvers and the fundamental differences among them. Second, it introduces the coupling of a new open-source linear solver capable of heterogeneous parallelism over multi-core central processing units and graphics processing units. Third, it compares linear solvers using a variety of mathematical programming problems, including standard test problems for linear and nonlinear optimization, optimal power flow benchmarks, and scalable two- and three-dimensional partial differential equation and optimal control problems. Finally, linear solver recommendations are provided to maximize Ipopt performance across different application domains.

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