Papers

This paper presents a data-driven method for modeling the pressure response of a hydraulic clutch control circuit. The system consists of a variable-force solenoid, accumulator, pressure regulator valve, and latch valve, and exhibits nonlinear behavior caused by hysteresis, latch transitions, and actuator dynamics. A baseline model using commanded current variables captured the general pressure response but failed to represent hysteresis and latch behavior accurately. The input vector was therefore extended with current derivative information, and several classifiers were tested to separate latch-related operating regimes before fitting Gaussian Process regression models to the resulting partitions. Nonlinear SVC and gradient boosting produced the highest latch-classification accuracy, and nonlinear SVC was selected for the final local-regression pipeline. The proposed approach was evaluated on unseen ramp-rate data and compared against a physics-based Amesim model. The machine-learning model reproduced the measured pressure response and hysteresis behavior more accurately than the physics-based simulation for the tested operating conditions. These results suggest that machine-learning plant models can complement physics-based hydraulic models during hardware development and controller calibration when representative test-stand data are available.

We study the Service-Aware Multi-Commodity Flow (SAMCF) problem, in which demand is elastic and governed by a logit choice model while routing is subject to hard capacity constraints. In a centralized, system-optimal setting, the network operator jointly determines how much demand to serve and how to route it. We formulate the SAMCF as a nonlinear program and propose an iterative fixed-point heuristic that alternates between solving an inelastic MCF via column generation and updating demand from the resulting service levels. Two linear approximations based on piecewise-linear demand functions and McCormick envelopes serve as benchmarks, while a piecewise-linear outer-approximation of the demand function is used to provide valid lower bounds. Computational experiments on public transport instances show that the heuristic finds near-optimal solutions in under two seconds - orders of magnitude faster than the benchmark methods - while matching their solution quality on all instances they can solve within a ten-minute time limit.

Input–output feedback linearization fails at decoupling singularities, where the decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded. This paper develops a singularity-robust trajectory-tracking controller for square nonlinear control-affine systems that tracks through isolated decoupling singularities with bounded control. The method recasts tracking as real-time arc-length homotopy continuation, equivalently a continuous-time Newton/Davidenko flow, and replaces the inverse decoupling matrix by the least-norm Moore–Penrose solution of an augmented matrix $A=[\Lambda\mid b]$, where $b$ is the homotopy direction. A transversality condition $w^T b \ne 0$, with $w$ in the left null space of the decoupling matrix, keeps the augmented matrix full row rank through a generic rank-one loss. The resulting flow agrees with feedback linearization away from the singular set, tracks with $O(1/k)$ error, and re-locks after each crossing. The theory also characterizes the reflection-versus-branch-crossing dichotomy at Whitney folds and relates the reflection case to a Filippov sliding mode. Extensions cover dynamic relative-degree-one minimum-phase systems and arbitrary relative degree via filtered-error reduction. The core results are formalized and machine-checked in the Lean proof assistant, including the multi-parameter (higher-corank) repair criterion, the uniqueness of the sliding motion at the fold, and the machine-checked treatment of Filippov solutions. Simulations include a redundant 2-DOF manipulator, relative-degree-one and relative-degree-two plants, and a dual-active-bridge series-resonant DC/DC converter, where the method performs bounded inversion across buck/boost and resonance singularities while preserving zero-voltage soft switching.

We study second-order optimality conditions for optimal control problems with integral cost. We consider extremals obtained by concatenating finitely many regular arcs and prove both a necessary condition for weak local optimality and a sufficient condition for strong local optimality within this class. The key object is the Jacobi curve, a curve of Lagrangian subspaces encoding the second variation along the reference extremal. In the regular case, this curve is smooth and can be identified with the tangent spaces to a suitable field of extremals. For piecewise regular extremals, the field of extremals is only piecewise smooth, and the associated Jacobi curve has discontinuities at the switching times. This makes the relation between vertical intersections, conjugate points, and optimality more delicate. We develop this discontinuous Jacobi-curve framework and show how it yields effective second-order tests. The case of one-dimensional free-control systems is studied in detail as an illustrative class in which piecewise regular extremals arise naturally.

In this paper, we develop a direction adaptation evolution strategy (DAES) – a new MAES-type method – for noisy derivative-free optimization, designed to reconcile the population-based search mechanisms of evolution strategies with rigorous complexity analysis. Unlike standard MAES schemes, DAES fixes the adaptation matrix to the identity and replaces matrix adaptation with a structured direction-generation mechanism based on symmetric sampling, joint sorting-selection of noisy function values, three-group recombination, and a new triangular search direction. Specifically, candidates are sampled along paired positive and negative directions, their inexact function values are jointly ranked, and the reordered directions are partitioned into three groups to construct three recombination points whose geometry defines the triangular direction. A signed sufficient-decrease search and extrapolation mechanism is then applied along this direction. This structure yields a population-based MAES-type algorithm that retains competitive practical behavior while being amenable to nonasymptotic analysis under noisy evaluations. We establish high-probability complexity bounds for nonconvex, convex, and strongly convex objective functions and derive corresponding guarantees at the noise-limited accuracy level. To the best of our knowledge, these results provide the first high-probability complexity guarantees for a noisy MAES-type derivative-free method with this direction-adaptation structure. Finally, numerical experiments on the 655 prince test problems from the BARON collection compare DAES with the advanced MAES-type solver MADFO and show a favorable trade-off between evaluation efficiency and ultimate robustness.

We study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton–Jacobi–Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.

Plug-and-Play (PnP) methods have emerged as a highly effective paradigm for solving imaging inverse problems by replacing traditional proximity operators of regularization terms with highly expressive deep denoisers. While empirically successful, establishing rigorous convergence guarantees for PnP algorithms remains a major challenge. Existing provable approaches based on the Gradient-Step (GS) denoiser suffer from theoretical and practical limitations, such as restrictive bounds on the regularization parameter, rigid step-size rules, and the inability to handle nonconvex data-fidelity terms. In this paper, we introduce PnP-IPA (Plug-and-Play Inexact Proximal Algorithm), a novel optimization scheme that overcomes these bottlenecks. We propose a new splitting strategy that evaluates the proximal operator of the scaled implicit regularizer inexactly. To enable adaptive step-size selection without exact objective evaluations, we design a novel surrogate merit function that successfully drives an Armijo-like backtracking line-search. Relying on the Kurdyka-Lojasiewicz property, we establish global convergence to a stationary point of the nonconvex objective without imposing any assumption on the regularization parameter. Extensive numerical experiments on image deblurring under both Gaussian and Cauchy noise demonstrate the practical advantages of PnP-IPA. By effectively lifting previous theoretical constraints, our method allows for optimal parameter tuning, yielding state-of-the-art restoration quality and robust convergence even in nonconvex regimes.

Nonautonomous saddle node bifurcations commonly appear in mathematical models, however, an optimal control theory tailored to such systems is not yet well established. We present conceptual work on the nonautonomous saddle node normal form. We develop an approach that reformulates the optimal control problem into an unconstrained minimization problem by exploiting a connection between bifurcation and rate-induced overshoots. The key component is a rate function that assigns the critical rate to every nonautonomous forcing. We show that in the saddle node normal form, the critical rate is unique, and hence the rate function is well defined. While this reformulation is valid in any case, the rate function is often unknown. To resolve this, we additionally present an approximation of the rate function that leads to a closed form expression of the approximated control. This makes the approach appealing in situations, in which we need repeated solves of the control problem.

Data-driven optimization often requires collecting data to estimate uncertain model parameters before solving the underlying decision problem. In practice, however, data acquisition may incur non-negligible costs, making it critical to determine when to stop additional data collection. In this paper, we study an optimal stopping problem for sequential data collection in stochastic optimization under parameter uncertainty. We propose a benefit-driven stopping framework that balances information gain and sampling cost. We model the unknown distribution parameter within a Bayesian learning framework and update beliefs sequentially as new observations are collected. At each iteration, the decision maker evaluates the expected marginal benefit of additional data relative to the unit sampling cost and determines whether to continue sampling or stop and implement the optimization decision. Based on this framework, we develop several stopping policies. The proposed policies are evaluated through a newsvendor problem with exponentially distributed demand. Numerical experiments compare the policies with fixed-budget and hindsight benchmark strategies. The results show that benefit-driven stopping rules can substantially reduce unnecessary data collection while achieving near-optimal decision performance, demonstrating the effectiveness of adaptive stopping in data-driven optimization.

Anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method, which reliably finds global minima of nonsmooth and nonconvex objective functions while being amenable to rigorous theoretical analysis, automatically detects and exploits additively separable structures of high-dimensional objective functions. This enables the algorithm to mitigate the curse of dimensionality where the objective function decomposes additively into lower-dimensional components. In this paper, we show this property proving that the computational complexity of anisotropic CBO depends exponentially only on the intrinsic dimension $\mathbf{d}$ of the objective function, rather than the ambient dimension $D\gg\mathbf{d}$. Additionally, we demonstrate that the computational complexity depends only on tractability conditions of the lower-dimensional components rather than on the full energy landscape, allowing for a more refined description of the objective function and algorithmic complexity, as the objective function landscape is captured directly at the level of the individual components. Our results highlight the effectiveness of anisotropic CBO for additively separable objective functions provided sufficient alignment between the structure of the anisotropic noise and the separability structure of the objective. This motivates the design of an enhanced algorithm that learns during optimization how to effectively explore the loss landscape by aligning the noise with the structure of the objective function, which we leave for future research. Numerical experiments validate our theoretical results, accentuating the influence of the intrinsic dimensionality $\mathbf{d}$, the level of separability, and the complexity and non-convexity of the objective within the separable components on performance and computational complexity of the anisotropic CBO algorithm.

Equitable renewable-energy planning is a sequential decision problem, but the decision variables available to a public planner differ sharply between mature and emerging economies. In the former the government largely builds generation, while in the latter it steers private investment through incentives and quotas. We formulate socially-equitable renewable-energy budget allocation as a Markov Decision Process (MDP) and, using a single problem-agnostic solver interface, compare the same policies across the two settings: eight U.S. cities (a mature economy) and West Java, Indonesia (an emerging economy). The results show that across both settings, a receding-horizon value-iteration policy dominates. In the U.S., it reaches 66% renewable penetration while cutting the underserved low-income population by 96% versus a random baseline. In West Java it closes the low-access gap while crowding in about seven times the private capital of a naive market-chasing heuristic. More interestingly, this heuristic, which is mildly sub-optimal in the U.S., could yield catastrophic outcomes in Indonesia, by underserving every low-access region, because chasing attractive markets and serving the underserved diverge once the planner acts through private developers.

We study recovery of a low-rank tensor $\mathcal{T}$ and a low-rank matrix $M$ from sparse, noisy observations. $\mathcal{T}$ and $M$ share one mode. We relax tensor rank using the nuclear norm of the mode-1 unfolding. This unfolding carries the coupling. It also has an exact proximal operator. We couple $\mathcal{T}$ and $M$ through a learned linear operator $G$. We prove a minimizer exists for the ridge-stabilized penalized objective. We prove that a proximal alternating linearized minimization (PALM) scheme converges to a critical point, for the algorithm as implemented, by verifying the hypotheses of a known nonconvex block-coordinate convergence theorem against our objective and identifying which conditions come from this problem's structure. For the matrix-only sub-problem, we state a proven sampling bound from matrix completion theory. For the coupled problem, we prove a sample-complexity result for a sequential sub-case: a separately-known coupling operator recovers $M$ from $\mathcal{T}$'s recovery accuracy alone, with no observations of $M$ needed. For the fully joint, alternately-estimated case, we state a conjecture and test it empirically, including a low-density regime where coupling does not help. We report multi-seed synthetic experiments with mean and standard deviation across sampling densities, an asymmetric-density experiment, and convergence curves, and we explain why recovery error stays high at low density. We apply the framework to workforce-skill and small-business-health estimation. Every application-specific choice is a proposed design, not a validated result; we have not run the framework on deployed data.

This paper presents an analytical approach to constructing the Pareto set in multiobjective variational control problems, based on Noether's second theorem. A fundamental connection is established between the gauge symmetries of the dynamical system describing the plant and the structure of the Pareto set. It is shown that the Pareto front can be interpreted as a conservation law arising from the invariance of the system and the quality criteria with respect to a symmetry group. A gauge regularization concept is proposed, which allows eliminating variables that do not affect the criteria without changing the Pareto set, thereby reducing the dimension of the solution space. Numerical examples are provided for the linear-quadratic regulator with two conflicting criteria and for the standard ZDT1 test problem.

Large-scale robotic swarms motivate the use of mean-field control (MFC). Classical partial differential equation (PDE)-based formulations provide a principled framework but can become computationally challenging in higher dimensions, whereas machine learning achieves scalability at the cost of approximation and guarantees. In this work, we establish an optimization-based framework that lifts the MFC problem into the space of occupation measures, resulting in a convex relaxation formulated as an optimization over measures. The resulting problem is solved using a Frank-Wolfe (FW) algorithm in the measure space, with each iteration reduced to a tractable optimal control problem. This approach retains the O(1/k) convergence rate of FW, avoids discretization of the state space, and naturally incorporates interaction and safety constraints. Numerical experiments demonstrate agreement with analytic and PDE-based baselines in two dimensions and show that the method scales to three-dimensional environments with multiple obstacles, where standard grid-based PDE solvers become impractical. A full 3D instance with ten obstacles is solved in minutes on a standard workstation, underscoring the practicality and scalability of the proposed framework.

The management of cash replenishment in Automated Teller Machine (ATM) networks requires scheduling recharges in order to minimize operational costs while maintaining high service levels and avoiding cash-outs, under uncertain and time-varying withdrawal demand. This work formulates the ATM cash replenishment problem through a Quadratic Unconstrained Binary Optimization (QUBO) model, which naturally captures nonlinear cost interactions, while incorporating operational constraints through penalty terms. The objective function combines fixed and variable replenishment costs with co-location discounts, as well as penalties for a late replenishment that could cause a service interruption. The resulting QUBO instances are solved using MegaQUBO, a GPU-accelerated QUBO solver. An empirical evaluation on a real dataset of 276 ATMs located in Italy, covering four representative months of 2022 (April, May, October, and November), benchmarks the proposed approach against a threshold-based operational policy. Results show consistent cost reductions of approximately 15%-18% while maintaining an excellent average service level (around 99.8%-99.9). Overall, the study demonstrates that QUBO-based optimization, coupled with GPU-based solving, can provide a practically deployable decision-support tool for large-scale ATM cash logistics.

This paper proposes a new notion of robust invertibility for nonlinear dynamical systems, and introduces constructive parameterizations of recurrent neural network which are robustly invertible by design. We define robust invertibility as the existence of a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input-output gains (the system is bi-Lipschitz), implying that both forward prediction and input reconstruction are robust to signal perturbations and initial-state mismatch. We construct robustly invertible recurrent models via series composition of static orthogonal layers and dynamic layers satisfying a strong input-output monotonicity property, and provide a differentiable neural network parameterizations in the form of the bi-Lipschitz recurrent equilibrium network (BiLipREN). Additionally, composition with dynamic orthogonal layers yields a nonlinear minimum-phase/all-pass (a.k.a. inner–outer) factorization. We illustrate the utility of the framework through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.

We consider bilevel optimization problems in which leader and follower jointly construct a feasible solution for an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage – or, in the pessimistic setting, enforce – a reaction of the follower that includes a set of mandatory items while excluding a set of forbidden items. Our investigation focuses on tractability results for various cases which emerge from different combinations of the total number of mandatory, forbidden, and neutral items. After providing some results for response problems that hold for any underlying combinatorial optimization problem, we examine response problems over the maximum-weight matching problem and the minimum-weight perfect matching problem as illustrative and surprisingly varied examples. Among other results, we show that the response problem is hard for even a single given mandatory or forbidden edge. On the other hand, it is fixed-parameter tractable with respect to the total number of non-mandatory edges. If, however, each follower's edge is either mandatory or forbidden, the response problem for the perfect matching problem is solvable in polynomial time while it remains NP-hard for the maximum-weight matching problem.

We study unconstrained bilinear zero-sum games, a fundamental model in online learning, adversarial optimization, and multi-agent decision-making. We introduce the implicit midpoint gradient descent rule, which we derive from continuous-time follow-the-regularized leader dynamics via symplectic integration methods. We prove that implicit midpoint gradient descent inherits several powerful properties from the continuous-time dynamics, including bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees. This is the first traditional online optimization approach to simultaneously achieve these properties in unconstrained bilinear zero-sum games. Finally, computational experiments demonstrate that the proposed method significantly outperforms the standard methods, optimistic and alternating gradient descent.

We study decentralized minimization of a sum of functions over a network where nodes may only leave, the remaining nodes stay connected, and the topology freezes between departures. Standard methods forget departed functions, causing a permanent bias proportional to the heterogeneity of the data. We propose Legacy Gradient Tracking (Legacy-GT): before leaving, a node compresses its function into a gradient-anchored quadratic legacy, bequeaths it to a neighbor, and provides a correction that exactly preserves the gradient-tracking invariant. We prove that the optimal legacy curvature is the average of the strong-convexity and smoothness constants, that legacies compose losslessly across chained departures, and that an adaptive anchor rule yields error bounds that decay geometrically after the last departure to a small residual–the minimum of the network's optimization error at departure and its heterogeneity radius. In contrast, the classic drop-and-forget baseline suffers a bias that never decays. Numerical experiments confirm the theory.

Heterogeneous mean-field control (MFC) involves multiple interacting populations with distinct dynamics, control constraints, and coupling structures. We develop a heterogeneous occupation-measure mean-field control (OM-MFC) framework that reformulates the population-distribution control problem as an optimization problem over occupation measures. In this representation, potentially nonlinear dynamics enter through linear weak Liouville constraints, while nonlocal inter-population interactions become structured bilinear functionals of the occupation-measure marginals. Exploiting this structure, we derive a Frank–Wolfe (FW) method and show that its linear minimization oracle decomposes into independent population-wise optimal control problems once the current population distributions are fixed, enabling parallel computation without an a priori discretization of the measure space. We further establish a sufficient convexity condition based on the symmetrized matrix-valued interaction kernel, under which Frank–Wolfe with an exact oracle admits the standard $O(1/k)$ convergence rate. Numerical examples on heterogeneous UAV coordination and three-dimensional search-and-rescue illustrate the framework both within the certified convex regime and for directional interactions beyond that regime.

Prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which limits their applicability. Motivated by this gap, we study prophet inequalities under two correlation models induced by a latent state of the world variable $Z$. In the common-base model, the algorithm observes the sequence $Z+X_1,\dots,Z+X_n$. We analyze single-threshold algorithms with the constraint that they always accept the final item, guaranteeing a reward of at least $Z$. When $Z$ is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of $0.381$. We then show that randomizing improves the guarantee to $0.4$. By a minimax argument, the same ratio is achievable when $Z$ is random. We depart from standard techniques by establishing a stronger lower bound of $0.41$ and an upper bound of $0.475$, ruling out the possibility that this class of algorithms attains the $1/2$ ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form and unifies known single-threshold prophet inequalities, yielding a simple threshold-optimality condition applicable to the common-base model. Finally, we study the common-scale model, where inputs take the form $Z\cdot X_1,\dots,Z\cdot X_n$. We show that this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding $1/n$.

The problem of multi-snapshot spike deconvolution is studied, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. A variable-projection formulation is adopted, in which the amplitudes are eliminated in closed form, thereby reducing the task to a nonconvex least-squares problem over the spike locations alone. This formulation is referred to as the variable-projection formulation of spike deconvolution (VarProSD). An explicit characterization of the basin of convexity of the VarProSD objective is provided in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation affect the local geometry. Within this basin, consistency of the estimator in the number of snapshots is established under stochastic noise, and a complementary, sharper error bound is derived under adversarial noise through the local Lipschitz property of the inverse map. Local convergence guarantees for gradient descent are further established when initialization is performed within the basin. A central role throughout the analysis is played by Beurling–Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments are presented to corroborate the theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.

Large-scale traffic assignment requires equilibrium models that are both behaviorally plausible and computationally tractable. This paper develops a perturbed utility Markovian equilibrium (PUME) framework that preserves the scalability of link-based Markovian traffic equilibrium models and extends their applicability to settings with boundary choice probabilities, undiscounted network loading, and general link interactions. As the behavioral basis of PUME, we first develop the perturbed utility Markovian choice model (PUMCM) in which the Bellman optimality operator is defined through a convex surplus function whose gradient directly yields the optimal policy. The model generalizes existing additive random utility (ARUM) Markovian choice models and admits both interior and boundary choice probabilities. Accordingly, unattractive links can receive zero flow without imposing ex ante choice-set restrictions as in existing ARUM models. We establish conditions under which the corresponding Markov decision problem is well posed and yields a proper demand mapping. We then formulate the equilibrium as a variational inequality (VI) problem on the dual cost space and establish its existence and uniqueness. Particularly, the VI formulation of PUME accommodates non-separable and asymmetric cost structures and thus offers a more flexible modeling framework than existing Markovian traffic equilibrium (MTE) models. For computation, we develop a modified policy iteration method for network loading and a safeguarded accelerated meta-algorithm for computing equilibrium. Both algorithms are proven to be globally convergent and have demonstrated satisfactory numerical performances. Experiments on benchmark and synthetic networks further show that the proposed framework is highly scalable and robust towards a wide variety of demand-supply settings.

Decision-making is posing an increasingly formidable challenge to investors because of the growing number of alternatives available in financial markets. A hot area of research over the past few decades has been portfolio optimization that seeks to determine how much an investor should invest in which asset. Introducing real-world conditions to the optimization model turns the problem into an NP-hard one for whose solution exact methods become inefficient; hence, researchers have turned to evolutionary algorithms to approximate solutions. In this paper, strengthening strategies are presented for multi-objective evolutionary algorithms that can provide a faster convergence rate and extensive search ability in the portfolio optimization problem under the cardinality constraint. To implement those features, a unique solution representation, a novel operator, and new repair mechanisms are introduced for solving the aforementioned problem in which lower and upper limits are set on the number of assets in the portfolio. For this purpose, new mating strategies along with the aforesaid package are implemented in well-known multi-objective evolutionary algorithms to solve the problem. The customized algorithms are subsequently tested against traditional ones using well-known market indices as benchmarks. Results indicate that the proposed strategy not only provides better approximations but also converges faster as well at no loss of performance with an increasing number of assets in the market.

Passive management has increasingly won popularity over the past few years because of its advantages, such as lower management fees and transaction costs. Index tracking endeavors to reproduce the performance of an index with smaller sets of assets. In this paper, a novel formulation is proposed that is not only more robust than the existing ones but also performs better on out-of-sample data and tracks indices over long periods without any considerable deviation or the need for rebalancing. Solving index tracking problems in a polynomial time is a challenging task due to their NP-hard nature. To address this issue, a novel heuristic based on metaheuristic algorithms and local branching is also developed to solve the proposed model. The heuristic enjoys not only the exploration capabilities of a genetic algorithm but the characteristics of local search algorithms as well. The data from the OR library is used to verify the capabilities of the proposed heuristic in comparison with commercial solvers. Results indicate that not only is the heuristic able to converge to optimal solutions for not-so-large problem sizes, but the portfolios it generates also outperform those yielded by commercial solvers in terms of both in-sample and out-of-sample data.

Kuznetsov and Sahinidis (J. Glob. Optim., 2025) prove distance bounds that confine optimal Lennard-Jones clusters and shrink the search region of deterministic solvers; their diameter bound charges each unit-width layer the loosest internal energy $-\binom{n}{2}$. We replace this by a certified estimate built from their own subset inequality and the proven minima $V_5^*$, $V_6^*$, and minimize the resulting layer bound over population profiles. Centred decreasing profiles supply candidate minima; an arrangement-free relaxation, whose only classical ingredient is the rearrangement inequality for sequences, closes every certificate over all profiles; and all comparisons are re-verified in directed-rounding arithmetic. For $5 \le N \le 200$ this certifies a one-layer improvement of the published bound at 92 sizes. The improvement resolves no open global-optimization case: it is a rigorous tightening of a published a priori bound, with a precise account of the mechanism. A direct downstream test on the only tractable sizes ($N \le 6$) finds the diameter box is not the binding resource for a deterministic solver there, and no solver runs at the sizes the refinement affects ($N \ge 38$); we therefore present the result as a theoretical note. We also derive the asymptotic form of the bound, $\rho_{KS} = N - \Theta(\sqrt{N})$, resolving a point left open by Kuznetsov and Sahinidis; the gain of the refinement itself grows like $\Theta(\sqrt{N})$ layers, with an exact asymptotic constant.

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

Multilevel optimization methods are highly effective for discretized energy minimization problems, but their Euclidean formulation does not directly apply to manifold constraints. We introduce a Riemannian extension of multilevel optimization based on a coarse model that is first-order coherent with the fine-level objective and yields descent directions under mild retraction-convexity assumptions. The framework includes metric-compatible vector transfer operators for passing first-order information between levels, covering both intrinsic and extrinsic constructions. We formulate two-level and multilevel algorithms and prove global convergence using a Riemannian Zoutendijk-type argument. Applications to Kohn–Sham density functional theory, Gross–Pitaevskii ground-state computation, and binary continuous cuts demonstrate the method on Stiefel, ellipsoid and Bernoulli manifolds. The experiments show significant reductions in computational time compared with single-level Riemannian optimization.

The increasing penetration of wind and solar resources into the power grid motivates the integration of flexible technologies to dynamically shift power loads in response to grid volatility and emergency events. The water electrolyzer presents a synergistic opportunity to provide flexibility through demand response (DR), while simultaneously electrifying hydrogen production; however, highly dynamic operation schedules accelerate device degradation. This work presents a mixed-integer linear program (MILP) optimization framework to study the multi-scale coupling between short-term operational flexibility provision in electrolysis devices and long-term stack replacement decisions driven by degradation. Active day-ahead market (DAM) participation of a 2.2 MW alkaline water electrolyzer over 22 years is solved as a case study. Our framework reveals that the multi-scale scheduling of DR operation and replacement decisions can extend optimal stack lifetimes by up to 2 years through load-shifting and further reduce lifetime electricity expenses by 33% relative to inflexible constant operation. Furthermore, we quantify key device parameter tradeoffs and next-generation design goals, where our analysis challenges the feasibility of the standard \$1/kg levelized cost of hydrogen (LCOH) production target solely through market arbitrage. Ultimately, this framework quantifies the largely unexploited economic value of multi-scale optimization in grid-integrated electrolysis.

In a real Hilbertian setting, we develop in this paper numerical splitting techniques guaranteeing strong convergence to the least norm solution of constrained variational inequalities. We develop a multiscale inertial forward-backward splitting algorithm for solving constrained monotone inclusion problems with multiscale penalization and vanishing Tikhonov regularization. The proposed framework accommodates smooth, nonsmooth, and mixed smooth–nonsmooth penalty operators, providing a unified treatment of a broad class of constrained monotone inclusion problems. In this general framework, we establish weak convergence of the generated iterates. By introducing a discrete Tikhonov central path, we further prove strong convergence to the minimum-norm solution of the problem under a mild constraint qualification condition on the problem data.