Papers

Interior-point (IP) methods are powerful second-order optimization algorithms but can make slow progress during early iterations. We propose Warm-IP, a hybrid framework that combines a path-following variant of the alternating direction method of multipliers (ADMM) with a primal–dual IP method. The ADMM phase generates a warm start that remains close to the central path, enabling the subsequent IP phase to rapidly refine the solution to high accuracy. We evaluate Warm-IP on a diverse collection of real-world and synthetic quadratic programming (QP) instances and compare it with leading commercial solvers (MOSEK and Gurobi) and open-source solvers (Clarabel, PIQP, and OSQP). We also curate and release a new benchmark of large-scale QPs from cancer radiotherapy treatment planning, built from real patient data and provided in canonical QP form, with instances containing hundreds of thousands of constraints. Warm-IP substantially outperforms existing solvers on large-scale problems. On the large-scale radiotherapy benchmark comprising 60 lung cancer cases, for example, Warm-IP solves an instance in 47 seconds on average with no failures, while the strongest competitors, MOSEK and Gurobi, average 55 seconds with failures on 23% of the cases and roughly 240 seconds with no failures, respectively. To support reproducibility and future benchmarking, we publicly release the implementation and benchmark data through a \href{https://github.com/Radiotherapy-Optimization/Warm-IP}{GitHub repository}. These results demonstrate that appropriately warm-started IP methods can combine the rapid initial progress of first-order methods with the high-accuracy convergence of second-order methods.

Variance reduction improves the convergence rate of stochastic non-convex optimization under the average smoothness condition. Existing adaptive variance reduction methods are especially designed for average smoothness and do not adapt to standard smoothness. In this paper, we study how one algorithm can adapt to these two settings and the noiseless setting without knowing the smoothness constants or the variance bound. Given the gradient budget $T$, we develop a two-phase method. The first phase compares two independent gradients and switches to normalized momentum when their difference exceeds a threshold. This phase provides both a candidate with standard smooth and noiseless guarantees and also ensures a bounded expected function gap for the second variance reduction phase. The resulting rates are $O(T^{-1/3})$ under average smoothness, $O(T^{-1/4})$ under standard smoothness, and $O(T^{-1/2})$ without noise. When $T$ is unknown, we propose a method with two paths that attain the corresponding rates with logarithmic factors in the stochastic settings. Here, we use a variance reduction path to deal with average smoothness and employ the AdaGrad-Norm for standard smoothness and noiseless setting. Neither method restarts from the initial point or requires a growing number of oracle calls in each round.

Momentum methods such as STORM and second-order momentum track the gradient by adding a transport correction to momentum, at the cost of an extra oracle call per iteration. We introduce randomized momentum transport, which evaluates the correction with probability $q$ and otherwise uses a cheap prediction, with importance weighting that preserves the conditional mean and adds $(q^{-1}-1)$ times the prediction loss to the variance. Under transport-bias, second-moment, and residual-control assumptions, linear minimization oracle (LMO) updates, including normalized, sign, and exact spectral (Muon) directions, retain an $O(T^{-1/3})$ stationarity bound for each fixed $q>0$ with a single-gradient initialization. For unnormalized STORM, some $q<1$ lowers the leading coefficient of our sufficient gradient budget if and only if the gradient-difference noise modulus exceeds the smoothness constant, by at most about $10\%$. An online diagonal predictor admits a regret-to-convergence guarantee. At approximately equal oracle cost, querying a quarter of the corrections lowers the final training loss by $18$ to $43\%$ on MNIST1D and raises the test accuracy of Hessian-vector transport on CIFAR-10 by $1.5$ points.

We present new algorithms for stochastic optimization problems in which the objective is neither smooth nor convex, subject to a convex constraint set. Our algorithms return a $(\delta,\varepsilon)$-Goldstein stationary point, as recently defined for constrained problems by Liu et al. (2024), with improved complexities, and are the first to achieve optimal convergence rates for this task in several settings. In particular, we design projection-based and projection-free algorithms that achieve a $\widetilde O(\delta^{-1}\varepsilon^{-3})$ rate using stochastic gradients, which we show to be optimal, and a $\widetilde O(d\delta^{-1}\varepsilon^{-3})$ rate using zeroth-order queries, which is also optimal in its dependence on the dimension $d$ and improves over prior work. Notably, our lower bound is given in terms of the Frank-Wolfe gap and holds even for smooth problems, which is of independent interest. These results show that in nonconvex stochastic optimization, the convergence rate is degraded neither by the presence of constraints nor by nonsmoothness. Our analysis is based on a reduction to online linear optimization, extending a related technique from unconstrained optimization.

Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms. We analyze a single-loop multi-objective method combining a norm-constrained linear minimization oracle with entropic updates of task weights. For any finite-dimensional norm pair, we establish an $O(T^{-1/2})$ ergodic stationarity rate in the deterministic setting, deriving the required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case. With stochastic gradients and a bounded-gradient assumption, one persistent blended-gradient momentum buffer yields an $O(T^{-1/4})$ rate whose leading term has no explicit dependence on the number of tasks. We also establish static regret for the inner weighting objective on arbitrary bounded gradient sequences and, with entropy regularization and a weight floor, iteration-wise inner-gap control after burn-in while retaining ergodic stationarity, conditional on a trajectory gradient bound. The optimal blended gradient can be nonunique in spectral and sign geometries, motivating a selection-independent inner gap. Our analysis extends the spectral construction studied by concurrent MOON to general norms. Experiments compare the method with existing multi-task optimizers and examine its oracle, weighting, and momentum choices.

We present a simulation-based method for the estimation, sensitivity analysis, and optimization of distortion risk measures (DRMs), a class of performance measures that capture a wide range of risk attitudes through an underlying distortion function. Existing simulation approaches often approximate DRMs using numerical partitions or empirical distributions. We show that, when the distortion function is piecewise polynomial, the nonlinear structure of a DRM can be handled exactly, without introducing distortion discretization error, using only simulated system outputs. This representation enables the development of a computational method that integrates finite-difference techniques for sensitivity analysis and for solving a broad class of differentiable risk-based optimization problems. Under suitable conditions, we establish strong consistency and convergence rates for DRM estimation, sensitivity analysis, and optimization. We further extend the approach to non-polynomial distortion functions through polynomial approximation. Numerical experiments on synthetic and calibrated financial examples demonstrate the effectiveness and robustness of the proposed approach across a range of applications.

This paper addresses the local convergence analysis of the stabilized sequential quadratic programming method for a class of composite optimization problems with convex piecewise linear- quadratic (CPLQ) modeling functions. We demonstrate that an appropriate second-order sufficient condition (SOSC) -formulated for composite problems- alone is sufficient to guarantee local superlin- ear convergence of the primal-dual sequence generated by this method. Furthermore, we examine the relaxation of the SOSC and show that it can be replaced with a weaker noncriticality of the associated Lagrange multiplier, provided, the modeling function is polyhedral and a relative interior condition is satisfied. We also present counterexamples showing that the relative interior condition is indispensable and that this relaxation fails for general CPLQ modeling functions.

Projection methods are among the most widely used techniques for solving convex feasibility problems due to their simplicity and robustness. However, classical projection-based algorithms and existing circumcentered-reflection methods generally provide only asymptotic convergence. In this paper, we introduce a relaxed version of the centralized circumcentered-reflection method (cCRM) obtained through an extrapolation of the circumcenter step. We first analyze the two-set case and then extend the proposed approach to the successive framework for multiple convex sets, including under- and over-relaxation and a backtracking strategy. The convergence analysis establishes Fejér monotonicity and global convergence under suitable conditions, while local error-bound assumptions yield linear convergence and standard regularity assumptions imply superlinear convergence. Motivated by finitely convergent projection methods based on perturbed feasible sets, we further investigate a perturbed relaxed formulation designed to achieve finite termination. Numerical experiments show good practical performance, comparable to and competitive with the finitely convergent PACA method. The results also provide clear numerical evidence of finite termination in the tested instances.

Fault diagnosis in asset-intensive industries requires multiclass models trained on noisy, sensitive data collected across geographically distributed client sites. Federated learning enables collaborative training without centralizing data, but most approaches do not provide formal privacy guarantees for individual records stored at client sites and can degrade under label noise. To address these limitations, we develop a differentially private, distributionally robust framework for federated multiclass classification that provides sample-level privacy for records stored at each client. We build on a federated Wasserstein distributionally robust multiclass objective with the Crammer-Singer loss, solved using an Alternating Direction Method of Multipliers (ADMM)-based algorithm. To protect the released client updates, our method injects calibrated Gaussian noise into the proximal linearized client updates of both the local model and the local distributionally robust variable before aggregation at the central server, and we quantify the cumulative privacy loss using the moments accountant. We provide per-round sample-level $(\epsilon,\delta)$-differential privacy guarantees and an overall composed privacy guarantee across all training rounds, together with convergence results and an explicit privacy-utility tradeoff. Experiments on simulated industrial fault data and UCI benchmark datasets show that our method outperforms differentially private federated empirical risk minimization and stochastic gradient descent baselines in most tested settings, with the largest gains under strong privacy-induced noise.

An inverse spectral problem for a $(1+1)$-dimensional wave equation with wave speed $c(x)$ is considered. The spectral data consist of the eigenvalues of the Neumann–Dirichlet problem for the operator $-c^2(x)\partial_x^2$ together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map $\Lambda$ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed $\Lambda$ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of $L^2$ and $H^1$ penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.

Recent results [arXiv:2602.04378, arXiv:2602.22608] have established worst-case objective-error lower bounds of order $1/T^2$ after $T$ linear minimization oracle (LMO) calls, matching the upper bound [arXiv:1406.1305] in its dependence on $T$ and thereby resolving a long-standing open question. Upon close inspection, however, both lower bounds are somewhat brittle. The first [arXiv:2602.04378] applies only to the standard Frank–Wolfe algorithm, and, as we show, its prescribed slow exact-line-search trajectories can amplify sufficiently small initialization errors exponentially, corresponding to roughly one additional bit of initialization accuracy per iteration. The second [arXiv:2602.22608] obtains lower bounds for any deterministic LMO-based first-order method via a resisting oracle that adversarially adapts to the queries of the algorithm. In both cases it is conceivable that randomization may circumvent these lower bounds; in particular, neither establishes a lower bound for unrestricted randomized first-order LMO methods. We close this gap by proving an $\Omega(L\operatorname{diam}(S)^2/T^2)$ distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. While our construction reuses the family of [arXiv:2602.22608], our proof characterizes the conditional distribution of the hidden permutation after adaptive queries and bounds the expected decrease of a combinatorial statistic controlling optimizer variance. We further show that the same error threshold holds with probability at least $3/4$ and yields a stopping-time tail bound.

We study the long-time behaviour of multi-dimensional first-order symmetrizable quasilinear hyperbolic systems with nonlinear feedback boundary conditions. Assuming strict interior and boundary Lyapunov inequalities for the linearized system and non-characteristic boundaries, we prove global existence for small initial data and exponential H^s decay, with s>d/2+1 an integer. We consider both bounded smooth domains and flat periodic cylinders, and illustrate the results with physical examples and numerical simulations.

We study a stationary ergodic mean-field game of scalar type where the representative player controls a diffusion through two-sided singular controls exerted only at the jump times of an independent Poisson process. For each mean-field parameter, we solve the novel constrained ergodic singular control problem, proving that the optimal policy is of threshold type: the diffusion is reset at Poisson arrival times to the nearest boundary point of an interval. Our explicit probabilistic derivation of the stationary distributions induced by Poissonian threshold singular controls is a key contribution of independent interest. This distribution is used to construct the mean-field fixed-point map. We prove existence and uniqueness of the stationary mean-field equilibrium by combining the stationary density with the monotonicity and continuity of the optimal thresholds in the mean-field parameter. The model provides a tractable framework for a case study of partially reversible investment with singular adjustments under exogenous intervention constraints.

Nonlinear model predictive control (NMPC) can be applied to large nonlinear systems with complex objectives while enforcing constraints, but the cost of solving the underlying optimization problems limits real-time application for systems with fast dynamics or many states. Neural networks can reduce this online cost by approximating the control law, but may violate constraints and typically require many precomputed optimizer solutions for training. The learning-based iterative solver for constrained optimization (LISCO) instead learns to predict primal-dual steps toward satisfaction of the Karush-Kuhn-Tucker (KKT) conditions through self-supervised training without precomputed solutions, but its dense, monolithic architecture grows with the prediction horizon. We remove this horizon dependence by exploiting the temporal structure of the problem in the solver architecture, which makes the network size independent of the horizon while leaving the training loss and the solver iteration unchanged. A closed-loop resampling strategy further aligns the training distribution with closed-loop operation at no additional sampling cost. On an economic NMPC benchmark of a CSTR cascade, the proposed solver scales to larger problems than monolithic LISCO and achieves median speedups of up to a factor of four over IPOPT. In closed loop, it achieves comparable control performance to IPOPT with no observed constraint violations.

Recovery risk measures incorporate explicit liability-recovery requirements into asset-liability risk assessment. We study their specialization to optimized certainty equivalents (OCEs), allowing the underlying utility specification to vary across recovery levels and encompassing Recovery Average Value at Risk (RecAV@R) as a special case. For finite recovery specifications, we use the dual penalty representation to characterize redundant recovery regimes and derive position-independent screening conditions. For a subfamily of the Cressie-Read power-divergence family, including entropic and truncated-quadratic specifications, we obtain an exact characterization on atomless probability spaces: universal redundancy is equivalent to a two-dimensional convex-hull condition involving recovery levels and divergence-scale parameters. On such spaces, universal redundancy for RecAV@R reduces to pairwise dominance, whereas for power-divergence OCEs it can arise from the joint effect of several recovery regimes. For nonlinear OCE utilities, we develop a certified piecewise-linear approximation framework for portfolio optimization. In finite-scenario models, the approximating problems are finite-dimensional linear programs, while utility approximation errors transfer directly to the Recovery OCE, yielding explicit error bounds and convergence rates. Uniform approximation over the portfolio domain further implies convergence of optimal values and stability of optimal solutions. A numerical asset-liability study illustrates the effects of recovery-dependent risk preferences and the accuracy and convergence of the approximation.

This paper presents a finite-time stabilization method for the integrator chain via optimal control techniques by introducing a tailored cost function. We analyze key properties of the resulting value function to derive the corresponding Hamilton-Jacobi-Bellman equation. Numerical simulations illustrate the effectiveness of the approach.

The open-loop minimum-energy control (MEC) problem asks for control inputs that minimize the control energy while ensuring the terminal state reaches a desired value. In this paper, we study the open-loop MEC problem for stochastic linear time-invariant systems. We consider a variety of stochastic constraints that enforce the terminal error to be small, and show that Stochastic MEC (SMEC) problem can be formulated as a QCQP. We obtain the optimal solution for SMEC and show that it generalizes the well-known Deterministic MEC problem by introducing novel stochastic variants of controllability and Gramian matrices. Finally, we show that there exists a fundamental lower bound on the optimal control energy which cannot be overcome by increasing the time horizon of the problem.

The paper is devoted to a broad class of optimal control problems governed by constrained sweeping processes with controlled dynamics, controlled prox-regular moving sets, and variable time included in the optimization process. Employing an advanced version of the method of discrete approximations, we verify its well-posedness with appropriate strong convergence of feasible and optimal solutions. This approach leads us to considering a parametric family of discrete-time controlled sweeping processes, which are reduced in turn to nondynamic mathematical programs with special types of geometric and functional constraints. Invoking powerful tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time and then—by passing to the limit over the diminishing step of discretization—continuous-time controlled sweeping processes. A crucial role in our device is played by explicit coderivative evaluations of set-valued mapping associated with the sweeping dynamics. In this way, we establish novel necessary optimality conditions for local minimizers of continuous-time controlled sweeping processes that contain absolutely continuous and BV adjoint arcs, local and global maximization conditions, $L^2$ active set multipliers, vector measures generated by sweeping and control constraints, and free-time transversality condition expressed through the averaged Hamiltonian. For suitably selected multipliers, we also obtain nondegenerate forms of these conditions when the optimal trajectory starts on the boundary of the moving set. The efficiency of the obtained results is illustrated by some examples related to practical modeling.

Timely information dissemination is essential in modern communication systems, where low informational latency is critical for networked control, monitoring, and real-time decision-making. The Age of Information (AoI) has emerged as a fundamental metric for quantifying information freshness at the receiver. A key challenge in AoI optimization is selecting the update generation rate: infrequent updates lead to stale information, whereas overly frequent updates increase queueing delays. We develop a structure-aware, data-driven framework for adaptive AoI optimization modeled as single-server queues with unknown system statistics. Our key idea is to exploit analytical representations of the AoI gradient, which depend on a finite-dimensional collection of system quantities that can be estimated from observations collected during system operation. This reduces the online learning problem to estimating the quantities required by the gradient, rather than learning the AoI objective as a black box. We propose an online stochastic approximation algorithm that jointly estimates these quantities and optimizes the update generation rate. We prove almost sure convergence of the algorithm to the optimal update generation rate under mild regularity and moment conditions, despite the temporal dependence induced by the underlying queueing process. We demonstrate the framework in four representative incomplete-information settings, including FCFS queues with unknown service characteristics or hidden background traffic, a preemptive LCFS system with partial observations, and a generate-at-will system with threshold waiting.

In the literature, research on representing the non-dominated set has focused almost exclusively on multi-objective optimization problems with Pareto-dominance. However, general ordering cones are used to incorporate preferences of decision makers into the optimization process and to model other specific types of optimization problems, such as ordinal optimization problems. In this work, we consider several representation approaches based on representation relations in the context of multi-objective optimization problems with polyhedral ordering cones. Our main research question is whether existing theory and solution approaches developed for the Pareto case can be used to address the more general setting.

We study pricing in a vehicle sharing network with $N$ locations and $C$ vehicles, motivated by on-demand bike, scooter, and car sharing, as well as robotaxi and ride sharing services. Customers request directed trips between locations, with travel times following general distributions, and leave without service if no vehicle is available. Pricing affects both demand for trips and the distribution of vehicles across locations. Computing an optimal dynamic pricing policy is challenging because the system state depends on the number of vehicles available at each location and in transit (including how long they have been traveling), motivating the use of simple static policies that offer a fixed price for each trip. For any number of vehicles and locations, we establish a tight performance guarantee for static pricing relative to optimal dynamic pricing. The guarantee holds for any concave objective functions (including revenue rate) and general travel time distributions. Our tight guarantee is bounded below by $ \frac{C}{C + \min \{ C -1, \sqrt{ CN} \} }$ (over 50%), achieving the first universal, non-asymptotic constant factor guarantee. We also prove that the optimal static policy can be efficiently computed by solving a convex optimization problem. Finally, we extend our performance guarantee to the setting with supply redirection.

We consider an optimal control problem with a general control constraint and a fixed initial state, for which convexity assumptions are absent. To establish the existence of optimal pairs, we introduce a control version of the Palais-Smale type condition adapted from critical point theory and prove that optimal pairs exist under this condition (see Theorem 2.1). Subsequently, we verify that a specific class of optimal control problems satisfies our Palais-Smale type condition (see Proposition 2.1). Furthermore, as an application, we present an example (Example 3.1) and show that, to the best of our knowledge, no existing result can be applied to establish the existence of an optimal pair in this case (see Section 3.1.1). Example 3.2 shows that finitely many Pontryagin maximum principle candidates do not guarantee the existence of an optimal control. Finally, Example 3.3 demonstrates that the proposed Palais-Smale type condition is sufficient, but not necessary, for the existence of optimal controls.

We study stochastic dynamic matching with abandonment, a fundamental model in which agents of different types arrive over time, may abandon the market while waiting, and can be matched in pairs for rewards that depend on their types. No online policy can guarantee more than $0.5$ of the omniscient benchmark, which knows all arrival and departure times. Whether this guarantee is achievable has been a central open problem, even in special cases. We resolve this problem by showing that, on every instance, a simple static priority policy, which is computable in polynomial time, earns at least $0.5$ of the omniscient benchmark. This establishes that the optimal competitive ratio is exactly $0.5$, improving the previous best guarantee of 0.192 for the general model [Li et al. 2026; Collina et al. 2020] Our analysis rests on a key stochastic result, which is a new lower bound on the probability that some agent from a given set of types is available. Building on this bound, we develop two approaches: a primal approach that reads the priorities from an optimal solution of a novel linear program, and a dual approach that assigns a “value” to each type, and prioritizes matches with high “surplus” (reward minus value of involved agents). These values are the unique solution of a fixed-point equation derived from linear programming duality. We also show that the two approaches are closely connected.

Conformal prediction provides finite-sample, distribution-free coverage guarantees that can support decision-making in specific settings. In multistage problems, earlier decisions determine the populations relevant to later calibration. When later scores are refit using earlier calibration results, reusing those cases for ranking need not preserve exchangeability with a surviving test case. We study sequential conformal calibration with a fresh fold at each stage: conditional on earlier folds, the selected fold scores and the surviving test score are exchangeable. This yields finite-sample coverage guarantees for every prefix of the decision sequence. For atomless survivor score distributions, we derive a universal Binomial-thinned Beta chain for the realized coverage process. Its law depends only on fold sizes and target levels, even under predictable refitting, and yields exact coverage slack, abstention probabilities, and calibration-conditional guarantees. We further characterize the tradeoff between freshness and data reuse. For fixed scores, strict-survivor reuse is exactly valid. For adaptive scores refit using past calibration data, reusing survivors can incur worst-case rank loss, while blocked designs provide an intermediate option between full reuse and stagewise fresh folds. As a case study, we apply the framework to serial multi-echelon inventory problem and propose a sequential conditional-quantile formulation. Each stock level is learned as a context-dependent quantile of cumulative demand among scenarios satisfying earlier no-backlog constraints. We connect these targets to common-shift stationarity and derive decision-level implications for the resulting calibrated policies. In a three-echelon simulation, fresh-fold refitting achieves 81.98% final-prefix coverage against an 81.82% target while reducing mean cost by 5.5% relative to the Bonferroni baseline.

Topology optimization typically involves a geometric volume/mass constraint due to the limited use of material. As a popular technique for topology optimization, the level-set method typically employs the Lagrange method, the penalty method, or the augmented Lagrangian approach to address the volume constraint. This requires careful parameter tuning and may lead to non-monotonic changes in the objective. In this paper, we present a new volume-controlled level-set method for topology optimization. By imposing a scalar weighted volume?compatibility constraint in a modified Stokes-type extension-regularization problem, the velocity is made volume compatible at the beginning of each frozen transport interval. At the continuous level, we prove that the initial rate of change of the regularized volume vanishes and that the subsequent frozen-velocity drift is of order $T^2$; under a narrow-band assumption, the corresponding constant is independent of the smoothing parameter. For the discontinuous Galerkin spatial semi-discretization of the level-set transport equation, we derive a quantitative regularized-volume drift estimate that displays the dependence on the smoothing parameter, spatial mesh size, transport time, and discrete velocity. The effectiveness of the present algorithm is demonstrated through the minimum compliance problem for linear elasticity, the eigenfrequency maximization for structural vibration, and the minimum energy dissipation problem for Stokes flows.

AI shopping assistants increasingly influence which products consumers consider. We study retail assortment planning under AI mediation, where a retailer chooses a catalog, an AI intermediary forms a customer-facing shortlist, and the customer chooses from that shortlist according to a multinomial logit model. We consider both deterministic mediation, in which the AI follows a fixed segment-specific ranking, and stochastic mediation based on the Plackett-Luce model. Empirically, we construct a dataset of product rankings from leading AI models, estimate Plackett-Luce ranking models, and evaluate their predictive performance on held-out assortments. Combining these estimates with synthetic customer preferences, we find that accounting for AI mediation can materially change optimal assortments and revenue. Across the tested capacity configurations, ignoring mediation yields mean revenue losses between 2.52% and 11.37% relative to the PL optimum. A much simpler deterministic plug-in, based only on the ordering of fitted AI utilities, performs substantially better, with mean losses of at most 0.53%. Motivated by this performance, we study assortment optimization under deterministic mediation. We characterize its computational complexity and show that, for any fixed number of customer segments, the problem admits a fully polynomial-time approximation scheme.

Spacecraft stationkeeping is subject to various operational uncertainties such as missed thrust events, maneuver execution errors, and dynamics modeling errors. Missed thrust events are the most disruptive of these, and current practices mitigate that risk through Monte Carlo-driven heuristics and conservative margins that offer no formal safety guarantees and do not scale to multiple missed thrust events. This paper models these unplanned thruster outages through the Markov Jump System framework, and represents maneuver execution errors and dynamics linearization errors as bounded multiplicative noise. The resulting closed-loop system is a linear time-varying periodic Markov jump system with multiplicative noise, for which Lyapunov-like mean square stability and periodic set invariance conditions are derived. A linear matrix inequality-based optimal controller synthesis framework then designs a periodic, mode-dependent mean-square stable feedback policy that respects path constraints with probability one. The framework is demonstrated on the Earth-Moon 9:2 near-rectilinear halo orbit (NRHO), under both impulsive and low-thrust propulsion, and on a Saturn-Enceladus $L_1$ NRHO using impulsive propulsion. Nonlinear Monte Carlo simulations show that the spacecraft recovers from an arbitrary number of missed thrust events while respecting path constraints, and that the multiplicative noise model introduces additional robustness to maneuver execution error and linearization error.

Mixed-integer programming (MIP) solvers are widely used, yet their performance can vary substantially even under seemingly neutral perturbations. Such variability complicates benchmarking, and the interpretation of computational studies. We introduce the Weighted Variability Score (WVS), a robust multi-component metric that aggregates dispersion in normalized runtime, final optimality gap, LP iterations, and branch-and-bound nodes across repeated runs. On a large subset of MIPLIB 2017 instances, each solved in 100 randomized runs under a four-hour limit, WVS reveals a right-skewed distribution with a substantial upper tail: many instances are essentially stable, but some exhibit large variability that is not explained by basic properties alone. Notably, this experiment proved optimality for the open instance 'neos-5045105-creuse' in two SCIP runs. Using static descriptors and root-level features, a tree-based predictor with a gated tail uplift achieves a mean absolute error of 0.058 and Spearman rank correlation of 0.65 on a sealed test split. Feature-importance analysis highlights root degeneracy, matrix structure, coefficient ranges, and presolve reductions as the main drivers of high WVS. Because variability is systematic and partially predictable from inexpensive early-run features, our findings offer a practical tool for preemptive tuning and lay a foundation for fairer evaluation of new algorithms.

Sobolev gradient flows for the Gross–Pitaevskii ground state problem are Riemannian gradient descent methods on the sphere under a metric induced by a discrete elliptic operator. We analyze their convergence through the condition number of the Riemannian Hessian at the ground state. Bounds of the Rayleigh quotient of the Riemannian Hessian give a uniform analysis of these methods for different metrics. The uniform analysis applies to the classical metrics, such as the shifted Laplacian and the energy-adaptive metrics. Inspired by this analysis, we also consider three new metrics. One of them is a Thomas–Fermi metric, whose condition number is bounded uniformly in the interaction strength for the one-dimensional harmonic trap and whose iteration count does not grow in the Thomas–Fermi regime in the numerical tests. The performance suggested by our analysis is consistent with three-dimensional numerical tests with up to one billion unknowns, for both Riemannian gradient descent and Riemannian conjugate gradient methods, whose practical performance is also related to the Hessian condition number. The Thomas–Fermi flow needs the fewest iterations, while the metrics inverted exactly by tensor-product solvers are the fastest in computational time on one GPU card.

Pseudospectral methods are a powerful framework for spacecraft trajectory optimisation that rely on fixed, problem-independent collocation nodes such as Legendre-Gauss-Lobatto (LGL) nodes. These nodes are located according to polynomial root distributions, which can result in insufficient resolution in regions of high dynamical activity, such as those encountered during collision avoidance manoeuvres. This paper presents an adaptive node distribution strategy for Birkhoff pseudospectral trajectory optimisation, where the collocation nodes are placed according to a probability density function derived from the thrust magnitude profile of the spacecraft. By concentrating nodes in regions of high dynamical variation, the proposed method simultaneously achieves feasibility preservation and improved convergence with fewer nodes in a collision avoidance case study when compared to the universal Birkhoff method, with Model Predictive Control serving as an independent validation comparator. The case study highlights the practical consequences of inter-node constraint violations and validates the adaptive node scheme as a robust and efficient alternative for fuel-optimal spacecraft manoeuvre design.