Papers

We study regularity properties of minimizers for the one-phase Alt–Caffarelli problem associated with the fractional $p$-Laplacian in the range $p\geq2$. We consider minimizers of the fractional $p$-energy penalized by the measure of the positivity set, with prescribed nonnegative exterior datum. We prove existence of minimizers and derive their basic variational properties: minimizers are nonnegative and are weak subsolutions of the homogeneous fractional $p$-Laplace equation. The main regularity argument combines fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, and a Campanato-type iteration. This yields local Hölder continuity of minimizers and implies that they solve the homogeneous equation in their positivity set. Finally, we prove the optimal free boundary growth estimate, showing that the sharp order known in the linear fractional Bernoulli problem persists in the fractional $p$-Laplacian setting.

The integration of distributed energy resources (DERs) into the power grid has introduced new challenges to AC optimal power flow (AC-OPF) problems. Traditional OPF optimize consider transmission systems, treating distribution networks as static loads. However, the growing presence of DERs makes accurate distribution system modeling crucial for grid operations. Consequently, efficiently solving the resultant large-scale, nonconvex transmission and distribution (T&D) AC-OPF problem remains a significant challenge. This paper proposes a Smoothed Two-Stage Decomposition Optimizer (StsDOpt) to address these complexities by decomposing the T&D AC-OPF problem into a master-subproblem(s) structure, enabling parallel solving. Unlike traditional methods, StsDOpt does not rely on approximations or relaxations. It uses a smoothing technique to render the subproblems responses differentiable with respect to the master problem, leveraging the barrier problem properties inherent in primal-dual interior point methods. This approach is crucial for accurately modeling and solving distribution systems, which are multiphase, unbalanced, and nonlinear, distinguishing StsDOpt apart from other methods. Integrated into the PowerModelsITD framework, StsDOpt has been validated through numerical experiments, demonstrating reduced wall-clock solve time and increased scalability. Results highlight its efficacy as a robust, scalable solution for large-scale T&D AC-OPF problems, facilitating the reliable integration of DERs into complex T&D systems.

For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

We study the problem of optimal control for dynamical systems near a pitchfork bifurcation, motivated by the role of external cues in guiding symmetry-breaking transitions in cell-fate selection and other natural processes. Using an asymptotic expansion of the optimality conditions obtained from the Pontryagin maximum principle, the leading-order optimal control law for a general n-dimensional system is examined across three dynamical regimes distinguished by scaling of control strength with respect to the distance from criticality. While in the strong control limit the results reduce to known approximations from linear-quadratic control, we derive generalized amplitude equations for the co-evolution of state and costate variables describing the optimized trajectory in the weak and intermediate control regimes. These control normal forms are validated against numerical solutions of the full optimal control problem for a canonical model of a bistable biochemical switch. The bifurcation structure of the optimal control problem is analyzed in the weak control regime. Finally, we demonstrate the construction of asymptotic solutions in the long time limit in this regime using boundary-layer methods.

Reinforcement learning (RL) has recently emerged as a promising feedback control strategy for nonlinear and complex dynamical systems. However, RL algorithms are sample inefficient and require a large number of interaction with the environment to synthesize optimal control strategies. Consequently, applications of RL are typically limited to sparse sensors and actuators due to the curse of dimensionality entailed by the exploration-exploitation dilemma in high-dimensional spaces. In this work, we bridge RL and traditional optimal control for dynamical system with a novel Physics-EnhAnced Reinforcement Learning (PEARL) paradigm tailored to the control of high-dimensional and parametric dynamical systems, exploiting the differentibility of their dynamics. Specifically, PEARL employs an actor-adjoint algorithm that leverages automatic differentiation to compute policy gradients over short horizons and adjoint-based sensitivities of future returns approximated via neural networks, significantly reducing the number of environment interactions, while mitigating long-term gradient instabilities. Through two challenging parametric navigation problems in unsteady flows, we show that PEARL (i) effectively exploits differentiable environments to outperform state-of-the-art RL algorithms, (ii) is sample efficient, thanks to the physics-guided policy learning, (iii) generalizes across multiple scenarios, which is crucial when dealing with parametric systems, and (iv) enables scaling RL to high-dimensional state and action spaces, without requiring low-dimensional state representations or multi-agent strategies.

We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provide a machine-checked Lean 4 formalization of the main result.

This paper illustrates the Pick-to-Learn methodology applied to the calibration of a Model Predictive Control policy. While developed around a specific example, the presentation is meant to highlight a methodology of broad applicability. The example concerns an aircraft traveling from an origin point to a destination point in the presence of uncertain crosswinds and a low-connectivity zone that should be avoided. The MPC policy is parameterized by two hyperparameters, which are selected from data by the P2L procedure. Starting from a dataset of 400 wind realizations, also called scenarios, P2L identifies a final compression set containing only two informative scenarios. The resulting MPC policy avoids the low-connectivity zone on all available scenarios and, according to the P2L theory, satisfies a probabilistic risk bound of $4.8\%$ at confidence level $1-10^{-5}$, where the risk is the probability of entering the low-connectivity zone in a future flight under a new wind realization not included in the sample.

The adaptive Bregman-Kaczmarz method recovers the exact, noise-free solution of a linear inverse problem even when every measurement it queries is corrupted, provided the corruption is fresh, independent and zero-mean. A block version was proposed for parallel hardware, but whether larger blocks actually converge faster was left open. We show the answer hinges on how the block is used: replacing the block sum with a block average collapses the analysis onto a single positive-semidefinite matrix, through which we prove that the guaranteed convergence improves monotonically with the batch size, with a total gain governed by the stable rank of the system matrix. We then address heterogeneous noise by introducing a noise-aware weighting that down-weights unreliable measurements, and prove it is strictly better than uniform weighting whenever the noise is not proportional to the row norms - a condition that essentially never holds in practice. The two improvements are compatible and their benefits combine. Finally, we show the adaptive step size interpolates automatically between a fast initial phase and a slowly vanishing tail that carries the error exactly to zero, and we explain how its hyperparameters can be estimated without knowing the true solution. Numerical experiments under sparse, heterogeneous noise illustrate these findings and confirm that the heuristic estimates produce effective step sizes.

We study the last iterate of the projected subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb{R}^d$. We prove that, for a finite horizon $n$ and a constant stepsize $\eta=\Theta(1/\sqrt n)$, the last iterate achieves an optimization error of order $d/\sqrt n$, showing that the extra $\log n$ factor appearing in high dimensions is unnecessary in every fixed dimension. We complement this result with a matching linear-in-$d$ lower bound and show that the sharp worst-case dimension-horizon dependence is of order $\min\{d,\log n\}/\sqrt n$. This solves, in particular, a COLT open problem posed by Koren and Segal in 2020 and shows that the correct dependence on the dimension is linear rather than logarithmic.

We propose a stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant (LTI) behavior with a Gaussian noise component. We show that this notion is a tractable subclass of stochastic behaviors and encompasses classical parametric stochastic LTI state-space system models as special cases. Analogously to deterministic LTI behaviors, the framework enables simple and tractable stochastic data-driven control methods. To this end, we obtain a method for prediction by conditioning the Gaussian behavior on the known part of the trajectory, which is identified directly from the sample covariance of trajectory data. Building on this method, we develop predictive control formulations that optimize over feedforward or disturbance affine feedback policies. The resulting formulations are shown to be convex. We further derive a finite-sample confidence bound on the prediction accounting for both aleatoric and epistemic uncertainty, and incorporate it into a robust control method, for which a tractable convex upper bound is obtained. Within this framework, subspace predictive control is recovered when only the mean prediction is used, while data-enabled predictive control is shown to account for the prediction uncertainty in an optimistic fashion. Numerical case studies illustrate the benefits of the proposed methods.

Urban transit electrification requires balancing energy efficiency, schedule reliability, and ride comfort for electric buses (EBs), particularly when interacting with transit-priority adaptive signals in congested networks. This paper proposes PriEco-DRL, a joint optimization framework that integrates EB eco-driving with transit-priority adaptive signal control using deep reinforcement learning (DRL). The signal layer employs a priority-weighted max-pressure (Priority-MP) controller to allocate green time based on occupancy-aware pressures, while the vehicle layer adapts longitudinal control based on uncertain and dynamically evolving local signal cues. A structured reward combines guidance and event-based reinforcement to align EB arrivals with green opportunities while considering energy, time, comfort, and safety. The framework uses centralized training and decentralized execution (CTDE) with parameter sharing, allowing a single DRL agent to learn from multiple buses and routes using local observations. Experiments on a real-world corridor show that PriEco-DRL reduces EB energy consumption while maintaining network efficiency and transit priority compared with fixed-time, actuated, and rule-based signal-vehicle coordination baselines. Energy- and trajectory-based analyses reveal that the improvements stem from fewer unscheduled stop-start events and smoother speed regulation under adaptive signals. The results highlight a tunable energy-time trade-off, allowing flexible operational choices through reward weighting.

The length-constrained cycle partition problem (LCCP) is a graph optimization problem in which a set of nodes must be partitioned into a minimum number of cycles. Every node is associated with a critical time and the length of every cycle must not exceed the critical time of any node in the cycle. We formulate LCCP as a set partitioning model and solve it using an exact branch-price-and-cut approach. Our dynamic programming-based pricing algorithm to generate improving cycles exploits the particular structure of the pricing problem for efficient bidirectional search and symmetry breaking. Computational results show that the LP relaxation of the set partitioning model produces very strong dual bounds and our branch-price-and-cut method improves significantly over the state of the art. It is able to solve previously solved instances in a fraction of the time and closes 14 previously unsolved instances with numerically safe bounds, one of which has 76 nodes, a notable improvement over the previous limit of 52 nodes.

We introduce a Monte-Carlo Tree Search solver for SAT and XSAT that pairs WalkSAT-style rollouts with a two-thread symmetry-breaking initialisation: one thread starts from all-true, the other from all-false, bounding each thread's initial Hamming distance to a satisfying assignment by $\lfloor n/2 \rfloor$. Empirically the solver closes 100/100 SATLIB graph-colouring encodings (flat200-479, sw100) in tens of milliseconds each, 20/20 planted 3-XOR-SAT at $n{=}200$ (median 10~s), 6/6 at $n{=}300$ (median 138~s), and one SAT Competition 2025 instance (Break-triple-04-06.xml.cnf) in 50~ms via the polarity split alone. On a full 16-round DES key-recovery encoding ($n{=}1976$, $m{=}30072$) it drives the negative-clause count from $\sim 200$ down to 24 (99.9% clauses satisfied) over 7 hours before hitting the S-box plateau.

We address the problem of optimally placing a limited number of sensors to reconstruct high-dimensional signals without knowledge of the underlying dynamics. The task is formulated as a nonconvex combinatorial optimisation problem and recast as a weakly convex constrained projection problem. This reformulation allows us to compute $\varepsilon$-global solutions using the Inexact Cutting Sphere algorithm. We further propose the Inverse Cutting Sphere algorithm, which starts from any feasible heuristic solution and either improves it by a prescribed tolerance $\varepsilon$ or certifies its $\varepsilon$-global optimality. The framework is evaluated on pressure reconstruction for NACA airfoils using XFOIL data.

We present frequency-domain models of a controlled mechanical system consisting of a flexible beam and a rigid body. The transfer function for the beam, governed by the Timoshenko equations under the body-beam interface conditions, is derived analytically. Based on this infinite-dimensional representation, reduced-order models are constructed within the framework of Loewner matrices. A comparative analysis of the Bode plots is presented for the beam model using different choices of sensor-actuator pairs to validate the proposed frequency-domain modeling framework.

This paper solves the targeting problem focusing on accuracy, computational efficiency, and reliability. The trajectory optimization problem is first recast as a polynomial optimization problem (POP) by leveraging differential algebra to compute high-order Taylor expansions of the nonlinear dynamics and constraints. Moment-sum-of-squares (SOS) optimization is then utilized to solve this POP. A convex formulation based on a second-order expansion of the dynamics is also proposed. For impulsive targeting, the moment-SOS and convex approaches are compared against traditional nonlinear programming (NLP) solvers and map inversion techniques. Results indicate that the moment-SOS approach provides solutions as accurate as traditional NLP, but with the critical advantage of guaranteeing convergence to the global optimum under mild assumptions. Furthermore, the method excels at handling large maneuvers and long propagation times, conditions in which standard linear approximations rapidly degrade. To demonstrate its versatility, the methodology is extended to a continuous low-thrust station keeping (SK) scenario in the Earth-Moon Circular Restricted Three-Body Problem. The algorithm's performance is then evaluated in the presence of significant state errors. The ability to directly handle non-convex constraints and recast complex, nonlinear dynamics into formulations with reliable convergence properties makes the moment-SOS approach suitable for autonomous onboard applications.

We consider the Burgers equation with a scalar control acting through a fixed spatial profile, in a regime where the linearized system is not controllable. We identify natural conditions on the control profile leading to quadratic obstructions to small-time local null-controllability, quantified by negative Sobolev norms of the control of arbitrary integer order. We show that these conditions are related to iterated Lie brackets. The proof relies on repeated integrations by parts in the quadratic expansion combined with nonlinear remainder estimates. Finally, we construct smooth spatial profiles that produce obstructions at every negative integer order.

Logic and optimization can, in combination, make valuable contributions to rule-based AI. Logic is the obvious medium for encoding a rule base and drawing inferences from it, while optimization provides a powerful technology for computing inferences. Their combination has taken on new relevance amid a growing concern for transparency in AI. which is important for reproducibility, explainability, trustworthiness, and fairness. Rule-based AI provides a natural solution to transparency that is becoming increasingly practical due to today's highly advanced optimization methods. This article surveys several areas of logic-optimization partnership, including probabilistic logic, Bayesian logic, belief logics and Dempster-Shafer theory, nonmonotonic (default) logic, many-valued logics, and inference of logical formulas from noisy data based on Boolean regression. It shows how to compute projections, the fundamental problem of both logic and optimization, using decision diagrams and logic-based Benders decomposition. It describes the use of postoptimality analysis to explain how conclusions are reached, further enhancing transparency, as well as the role of optimization in answer set programming modulo theories. The paper concludes by suggesting possible future research directions.

Derivative-free optimization (DFO) is challenging when the observation channel varies over time and evaluations are noisy. Conventional model-based methods assume stationary observations; under temporal drift, historical data biases gradient estimates and acceptance tests confound latent-objective decrease with temporal variation. We propose TOBYQA (Time-augmented Optimization BY Quadratic Approximation), a regularized model-based DFO framework that jointly incorporates spatial geometry and temporal variations within a single saddle-point interpolation system. TOBYQA augments the classical least-Frobenius-norm quadratic interpolation system with a linear-in-time drift term and a ridge regularization on the residual kernel, which accommodates noise and ensures well-posedness when the constraint block has full column rank, relaxing the geometric poisedness requirements of classical interpolation. We prove that under affine temporal drift, the recovered gradient is algebraically invariant to the drift rate for any noise scale and sample radius. This property leads to a drift-compensated acceptance test that subtracts the estimated temporal component from the observed reduction. Driven by an adaptive cubic regularization scheme with a closed-form step and a geometry-guarded statistical stationarity stopping rule, TOBYQA achieves an expected oracle complexity of $O(\varepsilon^{-2})$. Benchmark evaluations across diverse temporal drift regimes show that, at tolerance $\tau=10^{-3}$, TOBYQA solves 71.0%, 60.8%, and 41.9% of instances at $n=6$, $10$, and $20$, respectively, compared with 25.1%, 18.5%, and 14.9% for the best-performing comparison method. These results demonstrate higher solve rates under temporal drift while retaining comparable performance in static environments.

Wasserstein distributionally robust optimization (DRO) is commonly built around the empirical distribution, with the ambiguity radius selected from a concentration bound. Although this construction provides useful statistical guarantees, it can be conservative and does not fully exploit predictive information about the underlying distribution or the difficulty of a particular decision problem. We develop a more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius. The key requirement is not that the ambiguity set be centered at the empirical distribution, but that it contain the unknown data-generating distribution with the desired probability. We establish finite-sample guarantees and asymptotic consistency for arbitrary learned centers, derive tractable reformulations for non-uniform discrete predictive distributions, separate predictive-model and scenario-discretization errors, and prove stability under simultaneous perturbations of the center and radius. We further characterize the oracle conditional-quantile radius as the smallest conditionally valid rule and introduce a split-conformal procedure for finite-sample marginal calibration. Experiments on newsvendor problems, synthetic portfolios, distribution shifts, and real financial data show that learned and calibrated ambiguity sets can improve reliability, but do not automatically yield smaller radii or better decisions. Overall, the proposed framework treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.

Scheduling $n$ jobs with deadlines and processing times on a single resource to minimize late jobs equates to finding a maximum-cardinality feasible subset. Moore (1968) proposed a natural greedy algorithm for this: process jobs in nondecreasing order of processing time, adding each if the set remains feasible. While Zhao and Yuan recently matched the $O(n\log n)$ time of the classic Moore-Hodgson algorithm via amortized $O(\log n)$ feasibility queries, our first contribution is a simpler augmented BST data structure that achieves worst-case $O(\log n)$ query time. Moore's greedy algorithm is known to produce a maximum-cardinality feasible subset with minimum total processing time. Though the system of feasible subsets does not form a matroid, the algorithm behaves like a matroid greedy algorithm. Our second contribution explains this phenomenon: if all jobs have equal processing times, the problem corresponds to a nested matroid; with distinct processing time "tiers," Moore's algorithm effectively solves a sequence of nested matroid problems for each tier. Finally, we present an explicit linear-size flow network defining a polymatroid rank. A set is feasible exactly when its rank equals its total processing time, and Moore's algorithm accepts a job precisely when it yields its full processing-time marginal. By contracting shorter accepted jobs and measuring residual rank, we recover the tier matroid's complete rank function, offering a deep structural explanation for the greedy algorithm's correctness.

Fractional gradient descent (FGD) incorporates long-range memory through Caputo-type operators and has been shown to improve stability in ill-conditioned and nonconvex optimization problems. Despite these advantages, its practical use remains limited, mainly due to the high computational cost of evaluating history-dependent convolutions, which scales quadratically with the number of iterations. In this paper, we focus on making Caputo-based optimization computationally viable without sacrificing its intrinsic memory structure. We begin by expressing the fractional descent direction as a discrete convolution over past gradients, which provides a unified view of the method. Based on this formulation, we introduce two complementary mechanisms to reduce the cost of the memory term. The first uses a sum-of-exponentials (SOE) approximation of the power-law kernel, leading to efficient recursive updates. The second approach, newly proposed in this paper as dyadic hierarchical discrete convolution (DHDC), compresses the gradient history through a multiscale aggregation strategy. Rather than treating these approximations as purely numerical accelerations, we interpret them as perturbations of the ideal Caputo operator. This viewpoint allows us to analyze how the compressed memory affects the optimization dynamics. Under standard $\mu$-strong convexity and $L$-smoothness assumptions, we show that the resulting method still exhibits monotone descent and linear convergence, provided that the approximation error remains controlled.

We extend the parametric mean-payoff game framework of Parsons et al. to bi-objective tropical pseudolinear optimization with general two-sided constraints. The problem is to simultaneously minimize two tropical pseudolinear objectives over the feasible set of a general two-sided system U otimes x oplus b is less than or equal to V otimes x oplus d, we characterize the Pareto front via a parametric mean-payoff game in two parameters ( lambda 1, lambda 2). The feasibility region R is convex and the Pareto front P is a convex piecewise-linear curve with finitely many breakpoints, these properties are natural extensions of the single-parameter case to two parameters. In addition, we give as a new result, the joint denominator bound: the cycle coefficients satisfy k 1( gamma ) + k 2( gamma ) is less than or equal to 2 for any elementary cycle gamma, yielding | Delta | is less than or equal to 2 for every 2 times 2 Newton system, except in the fully decoupled case, and implying that all breakpoints have half-integer coordinates for integer data. Optimality and infeasibility certificates are given in terms of the cycle structure of the parametric game. Two algorithms are developed, a directional bisection algorithm (O(n squared (n+m) log M) per direction) and a Newton scheme tracing the complete Pareto front via 2 times 2 linear solves in at most | S | steps, independent of M. The directional bisection algorithm is pseudo-polynomial in n, m and M. The Newton scheme is independent of M but requires up to |S| steps, where |S| is exponential in n. Lastly, we give numerical experiments on random instances to confirm the directional bisection complexity bound exactly; the Newton scheme's worst-case bound is not attained by random instances but is shown to be tight via explicit adversarial constructions.

We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a unified variational framework that connects several equivalent formulations of the induced transport distance, including Lagrangian, Eulerian, convex optimization, Hamilton - Jacobi dual, and Hamiltonian flow formulations. Under standard convexity assumptions on the Lagrangian, we establish the equivalence of these formulations through variational arguments and convex duality. The resulting optimality system reveals a natural Hamiltonian structure on the Wasserstein space, providing a direct link between optimal transport, Hamiltonian dynamics, and optimal control. The proposed framework extends the classical quadratic-cost theory to general Lagrangian costs and offers a unified perspective for the analysis of transport metrics generated by action functionals.

Optimization models are built in a variety of modeling languages and solved by a variety of solvers, but once a solution exists, the information needed to understand it is fragmented: each solver exposes a partial, differently named set of native diagnostics, and the modeling language has already canonicalized the formulation the user wrote. We present pyoptexplain, a practitioner-first Python library for post-optimality analysis of optimization models that sits above this layer. It adapts a model authored in any of five modeling front ends, namely cvxpy, Pyomo, gurobipy, docplex, and OR-Tools, into a normalized internal representation, solves it through a choice of backends, and answers the why and what-if questions of an optimization decision through one uniform interface. The design rests on two observations. First, a post-optimality quantity requested from different backends for the same problem can come back as an exception, a structurally meaningless zero or a basis-dependent value that disagrees across solvers, so reporting whatever one solver returns is unreliable. pyoptexplain reports a quantity only when both the representation and the chosen backend can justify it, and does not approximate unavailable information. Second, repeated scenario analysis can amortize its cost by extracting the model once and reusing a warm solver session across a batch of scenarios. pyoptexplain builds a single scalable what-if interface, uniform across its modeling languages and backends and returning a certified report for every scenario, at a cost within a small constant factor of the bare solver. A reproducible computational study substantiates both claims. Source code is available at https://github.com/h-fellahi/pyoptexplain and installation can be done through the Python Package Index https://pypi.org/project/pyoptexplain/.

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

Emergency department (ED) boarding occurs when admitted patients remain in the ED while awaiting inpatient beds. Boarding is a major driver of ED crowding and has been associated with poor patient outcomes. We propose a framework to help EDs reduce boarding time and length of stay by using information about current patients and bed availability to proactively request inpatient beds before admission decisions are finalized. We formulate the problem as a Markov decision process in which predictions of each patient's admission probability and time to disposition are aggregated to guide early inpatient bed requests. This formulation leads to three data-driven policies based on approximate dynamic programming, reinforcement learning, and a newsvendor-type approach. Using a simulation model based on data from a large ED, we evaluate these policies across a wide range of settings. The simulation study shows that proactive aggregate bed requests can reduce average boarding times for admitted patients by 30-70% and average length of stay for all ED patients by 6-15%, while creating only modest idle time for prepared inpatient beds. The newsvendor heuristic provides the most attractive tradeoff between ED performance and inpatient bed idle time, whereas the reinforcement learning heuristic produces smoother bed-request patterns when stability in downstream hospital processes is especially important. Our work shows how EDs can use prediction tools to make proactive bed-request decisions that improve ED operations while helping managers balance reductions in ED delays against inpatient bed idle time. Our findings also illustrate the value of evaluating both simple myopic heuristics and more sophisticated reinforcement learning-based approaches, since each can offer distinct advantages depending on the performance measures and implementation constraints most important to managers.

This paper presents a proof-of-concept workflow for equation-oriented process optimization that runs entirely on a GPU. Process optimization models often incorporate complex interconnected unit operations, dynamics, and uncertainties, resulting in large nonlinear programs that can be computationally demanding for conventional CPU-based solvers. Although emerging GPU-based solvers offer substantial computational benefits, their application to process optimization has been limited by the lack of GPU-compatible process modeling tools. We address this gap by prototyping the GPU-compatible process optimization models using an existing GPU-capable optimization software stack, including ExaModels (algebraic modeling system), MadNLP (optimization solver), and cuDSS (linear solver). ExaModels formulates the process optimization problem in a GPU-compatible way by exposing its repeated algebraic structure, while MadNLP and cuDSS solve the resulting nonlinear program on the GPU. This workflow is demonstrated on a CO2 absorber design problem under feed uncertainty, in which a shared column diameter is minimized subject to equilibrium and hydraulic constraints in all scenarios. For the largest case with 5,000 scenarios and 1.5 million variables, the GPU workflow achieves a speedup of approximately 21\times over a single-threaded CPU baseline using JuMP, Ipopt, and MA57.

Multi-objective learning (MOL) aims to optimize multiple objectives simultaneously. The multi-gradient descent algorithm (MGDA) is a workhorse that iteratively updates along a common descent or conflict-avoidant (CA) direction across objectives. In stochastic settings, however, the vanilla stochastic MGDA method, SMG, lacks a fast convergence rate because mini-batch sampling introduces noise in the gradients. This causes bias in the update direction, which is controlled by the CA direction continuity. In this paper, we show that the CA direction is $1/2$-Holder continuous with respect to the Jacobian matrix, and the exponent $1/2$ cannot be improved in the worst case. This leads to a suboptimal convergence rate for vanilla stochastic MGDA in prior works. Nevertheless, under additional regularity conditions, we show this can be improved to Lipschitz continuity. Based on this insight, we propose a stochastic multi-objective regularity-aware (MoRe) method that exploits the Lipschitz continuity of the CA direction when the subproblem is regular, and switches to a fixed scalarization weight otherwise. Intuitively, the proposed algorithm employs CA direction update when the gradient conflict is large, and linear scalarization update otherwise. Theoretically, our method improves the convergence rate of SMG in the nonconvex setting from $\widetilde{\mathcal O}(T^{-1/4})$ to $\widetilde{\mathcal O}(T^{-1/2})$, where $\widetilde{\mathcal O}(\cdot)$ hides logarithmic factors. Meanwhile, we also establish the per-iterate conflict-avoidance guarantees. Empirically, experiments demonstrate its effectiveness in multi-task performance and verify convergence behavior consistent with the established theoretical rate.

We investigate the computational complexity of bilevel integer linear programming. While Jeroslow~(1985) established that the decision version of this problem is $\Sigma^p_2$-complete when restricted to binary variables, we prove that this $\Sigma^p_2$-completeness persists even for general integer variables, settling a question that remained open for over 40 years. Furthermore, we analyze the impact of various structural assumptions on computational complexity. Notably, we strengthen the result of Köppe et al.~(2010) by proving polynomial-time solvability whenever the total number of upper- and lower-level variables is fixed, without any additional assumptions.