Papers

We propose to learn the unknown dynamics in the proximity of a small body directly from tracking data, representing them as a feed-forward neural network embedded in the system Hamiltonian. The equations of motion form a Neural Hamiltonian Ordinary Differential Equation, whose variational equations provide exact training gradients: estimation uses position and velocity arcs at realistic noise levels, without acceleration or potential labels, and a continual learning approach warm-starts the network as new data are acquired. The known part of the Hamiltonian carries whatever is available, from the central term and spin state to the constant-density model of the imaged shape. We assess the method against a normalized spherical harmonics expansion estimated from identical arcs through the same machinery, on scenarios built on the shapes of Itokawa, 67P, Bennu and Eros. The network remains usable inside the Brillouin sphere: it plans ballistic descents at Itokawa to \SI{4.6}{m} median touchdown error from tracking alone, against 5.1–\SI{48.9}{m} for harmonics of degree 4–12, and to \SI{0.9}{m} with the imaged shape as prior, a configuration that also recovers localised density anomalies invisible to any harmonics degree. The two representations are complementary, and we discuss their combined use across the phases of a small-body mission.

Lung cancer remains a leading cause of cancer mortality because many cases are diagnosed at advanced stages. Low-dose computed tomography (LDCT) screening can reduce mortality through earlier detection. Partially observable Markov decision process (POMDP) models can personalize screening by maintaining a belief over an individual's latent cancer state. However, cancer-state transition probabilities are often generated from clinical simulations and are subject to estimation error and model misspecification. We propose a robust POMDP framework using $\ell_1$-norm ambiguity sets around the nominal transition probability. The model optimizes screening decisions against the worst-case transition probability while keeping other components fixed at nominal values. Building on the piecewise-linear and convex structure of the robust value function, we adapt point-based value iteration to compute robust screening policies. We evaluate the policies using out-of-sample simulations that perturb selected cancer-progression parameters and compare them with the nominal ENGAGE policy for representative female and male heavy-smoker cohorts at age 50. Robust POMDP policies generally outperform nominal ENGAGE in mean out-of-sample quality-adjusted life-years (QALYs), with the best performance at a moderate ambiguity radius within the tested grid. Clinical analysis shows that the robust policy reduces lung cancer deaths (LCDs) in all evaluated settings for the female cohort and in most settings for the male cohort, with additional false positives (FPs). Screening-schedule analysis shows that the robust policy recommends more LDCT screens and detects more early-stage lung cancers. These findings show that incorporating transition-model uncertainty into data-driven screening models can improve out-of-sample reliability and provide more robust decision support when clinical simulation inputs are misspecified.

We introduce AdamX, a first-order optimizer that incorporates cosine similarity as an adaptive mechanism for controlling update magnitudes. The proposed method is scalable, model-agnostic, and straightforward to integrate into existing training pipelines. We further introduce a variance rectification scheme that promotes smoother optimization during the early stages of training. Overall, we provide empirical evidence that AdamX achieves competitive convergence rates across a range of benchmark datasets and architectures. Performance is evaluated in terms of the number of epochs required to reach predefined performance thresholds under a fixed hyperparameter budget. Code and Experiments available at: https://github.com/FranciscoCaldas/adamX.

We study strong consistency and optimal tracking of the Astrom-Wittenmark self-tuning regulator with unknown input gain. Existing results establish full parameter consistency for the unmodified recursion under additional growth conditions on reference information. Other approaches obtain performance guarantees by adjusting the estimates used in feedback or adding decaying probing signals. For a class of minimum-phase linear systems with martingale difference noise, we prove that ordinary least squares with certainty-equivalent control achieves stability and optimal tracking almost surely for each bounded reference, without reference excitation, gain adjustment or added probing. When at least two parameters are estimated, all parameter estimates are strongly consistent, with squared estimation error norm $O(\log\log\log n/\log n)$ almost surely. An exact law of the iterated logarithm under zero reference shows that this rate cannot in general be improved. We also derive a logarithm law for cumulative squared tracking residuals that makes their dependence on reference energy explicit. A chi-square limit yields asymptotically exact parameter confidence ellipsoids for each bounded reference. For consistency, we prove a logarithmic lower bound on the cumulative squared difference between an auxiliary least-squares prediction of the noise and the reference. If the gain error persisted, this bound and the actual least-squares recursion would give incompatible bounds on the same weighted squared sum. This contradiction establishes gain consistency before stability.

Increasingly frequent and severe wildfires threaten ecosystems, public health, and infrastructure. Early detection is vital but limited by existing monitoring systems. Drones offer mobile, real-time coverage, but optimizing sensor placement and drone routing in dynamic fire zones remains challenging. To address this, we introduce WFDroneBench, an open-source Python benchmarking library for early wildfire detection that integrates machine-learned risk maps with optimization-based deployment strategies for sensors, charging stations, and drones. It evaluates risk maps, optimization strategies, and monitoring equipment using standardized metrics and realistic wildfire simulations. The framework supports benchmarking across predictive and decision-making components: machine learning researchers can assess risk models and compare routing strategies. WFDroneBench includes 7746 scenarios across 49 locations, built from historical ignitions, real-world wildfire risk maps, and simulated fire spread, along with two ground detector and three drone routing strategies. Our experiments show that the risk-aware strategy Max-Coverage significantly outperforms other baselines when risk maps are sufficiently accurate, achieving the fastest detection on the most difficult fires. We further find that risk-aware static infrastructure helps even under an imperfect risk map and drone-based detection outperforms ground sensors. Finally, our results reveal two key open challenges: (i) detecting small fires rapidly and reliably, and (ii) improving risk-map prediction, where the gap between ground-truth ignition patterns and available risk maps highlights a significant opportunity for ML innovation. We openly release all code, data, and documentation.

This article presents a data-driven approach to minimally adjust forecast-based steady-state generation dispatch in electrical transmission networks. The goal is to manage congestion caused by uncertainty in demand and renewable generation under a fixed automatic generation control (AGC) policy operating without saturation. Under this policy, we first show that the post-AGC residual injection distribution is invariant to the scheduled generation dispatch. This result separates line loading into a dispatch-dependent baseline and an uncertainty-induced residual component. Using historical forecast errors, we then apply split conformal prediction to construct transmission-line margins with distribution-free, finite-sample linewise or joint coverage guarantees. We also calibrate the aggregate pre-AGC mismatch to determine the generation headroom required for the prescribed AGC response. We incorporate these calibrated quantities into a convex secure re-dispatch problem that minimally adjusts the nominal schedule while reserving sufficient transmission capacity and AGC headroom. The resulting corrected dispatch provides finite-sample probabilistic guarantees on per-line security and unsaturated AGC operation. Case studies on the RTS-GMLC system demonstrate substantial reductions in out-of-sample line-limit violations with modest operating-cost increases.

We study multi-product replenishment subject to aggregate operating limits imposed pathwise in place of expectation. On-hand and in-transit inventory, warehouse capacity and an emission allowance define a safe set, and the resulting multi-product lost-sales problem is posed as a safe stochastic optimal control problem. The Hamilton–Jacobi–Bellman (HJB) equation is represented as a system of forward–backward stochastic differential equations (FBSDEs) and solved by learning the value function and its gradient, with a quadratic program derived by control barrier function (CBF) is solved at each decision epoch. Because purchase orders enter the in-transit pipeline only, the in-transit constraints have relative degree one while the on-hand constraints have relative degree two. We extend the stochastic high-relative-degree construction of (Clark, 2021) from a single barrier to four coupled aggregate constraints of mixed relative degree, and we establish its initial-condition requirement by design.

Predetermined stepsize schedules featuring carefully chosen long steps have recently been shown to accelerate gradient descent (GD) on smooth convex functions. A prominent class of such schedules is built through recursive composition. In this paper, we characterize the convergence of these optimized recursive schedules, revealing a nonconstant log-periodic modulation across prescribed horizons. Specifically, we prove that for every $N \geq 1$, the optimized ConPP/OBS-S schedules satisfy $$f(x_{N-1})-f_\ast \le \frac{1}{2N^p\Phi(\log_2N)-1} \frac{L}{2}\|x_0-x_\ast\|^2, \qquad p=\log_2(1+\sqrt2), $$ where $\Phi$ is a positive, Lipschitz, nonconstant $1$-periodic function. We derive this by proving that balanced splitting is optimal at every horizon for these constructions, resolving a conjecture of Zhang and Jiang. Furthermore, for their asymmetric counterparts (ConPD/OBS-F schedules), we show that although optimal splits are not necessarily balanced, the same Silver exponent asymptotically persists alongside a distinct log-periodic modulation.

Grimmer, Shu, and Wang asked whether a balanced constant schedule is $s$-composable at every horizon. More precisely, for an integer $n\geq 1$, let $\bar{h}=1+r$, where $r\in(0,1)$ is the unique solution of $$r^n\bigl(1+n(1+r)\bigr)=1,$$ and run gradient descent for $n$ steps with normalized stepsize $\bar{h}$. The cases $n=1,2$ were known, while the general case $n\geq3$ was left open. We prove the conjecture for every $n$. Our proof gives an explicit, dimension-free nonnegative linear combination of the smooth convex interpolation inequalities. The certificate is assembled from matrices supported on contiguous index intervals. Its off-diagonal entries are automatically positive, its quadratic part is diagonal, and its remaining multipliers reduce to two scalar families. We derive closed forms for those families and prove positivity using strict concavity and elementary rational inequalities. Consequently, for every $L$-smooth convex function, the constant schedule satisfies the sharp mixed terminal Lyapunov inequality conjectured in the original paper, together with the associated simultaneous objective-gap and gradient-norm bounds. All exceptional horizons and boundary indices are treated explicitly.

This paper introduces Stochastic Online Scaled Gradient Methods (SOSGM), a generalization of the recently developed adaptive preconditioning framework in arXiv:2505.23081 and arXiv:2509.11007 to stochastic optimization. Under standard assumptions, we establish convergence guarantees for SOSGM using large batchsize or variance reduction. SOSGM is compatible with popular diagonal and/or low-rank preconditioners as well as heavy-ball momentum, while maintaining memory and computation cost comparable to Adam. Extensive numerical experiments demonstrate the strong empirical performance of SOSGM. Using a diagonal preconditioner, SOSGM and its variants substantially outperform existing adaptive first-order methods across a range of statistical learning tasks.

We investigate mean-variance portfolio selection with an $\ell_0$-penalty to promote sparsity in asset allocations. Uncertainty in the mean return vector is incorporated through an ellipsoidal uncertainty set, yielding a robust sparse optimization framework. We characterize the structure of both local and global minimizers and exploit these properties in the risk minimization and return maximization formulations. Building on this structural insight, we develop a branch-and-bound algorithm tailored to the resulting robust sparse portfolio problems, together with a new pruning rule that can discard exponentially many candidate portfolios in a single step. Extensive computational experiments on real market data, together with comparisons against a mixed-integer second-order cone programming solver, demonstrate the effectiveness and competitiveness of the proposed approach.

Linear programming plays a fundamental role in computer science, with applications in optimization, formal verification, SMT solving, and numerous other domains. When exactness guarantees are required, numerical inaccuracies arising from floating-point arithmetic can compromise soundness, whereas exact rational arithmetic often incurs significant computational overhead. In this paper, we investigate the numerical stability of the simplex algorithm and establish conditions under which a precision-boosting floating-point implementation provably produces the same pivot decisions and final basis as an exact rational implementation, from which the final result is reconstructed and certified exactly. Our analysis shows that correctness depends on both arithmetic precision and a careful handling of numerical tolerances. Based on these results, we develop a tolerance-aware, precision-boosting simplex algorithm with formal correctness guarantees. Finally, we introduce a delta-complete termination criterion that allows the algorithm to terminate once certified upper and lower bounds on the optimal objective differ by at most a user-specified threshold delta, providing a certified, user-controlled optimality gap.

While existing strategies for Flight and Maintenance Planning for the defense sector generally address idealized conditions, real-world planning often involve non-nominal initial fleet states and complex inspection schemes. To address these challenges, we propose a multi-year planning strategy that maximizes long-term fleet availability. The formulation incorporates multiple competing objectives capturing the tight coupling between aircraft usage and maintenance, while enforcing cyclic inspection requirements and limited maintenance-dock capacity. To enhance practical applicability, we introduce three complementary strategies: a receding-horizon approach to reduce computational burden, a Bayesian Optimization routine to automatically tune cost-function coefficients, and an aircraft-specific weighting scheme that adapts model priorities based on fleet-condition statistics. Results on three realistic benchmark scenarios demonstrate the effectiveness of the approach.

Many classes of two-player zero-sum stochastic games have the orderfield property: if all payoffs and transition probabilities lie in a subfield of $\mathbb{R}$, so does the undiscounted value. Absorbing games fail this property, and Oliu-Barton and Vigeral [Absorbing games with irrational values, Oper. Res. Lett. 51 (2023) 555–559] conjectured the precise extent of the failure: every real algebraic number $\alpha$ of degree $m\geq 1$ over $\mathbb{Q}$ is the undiscounted value of a rational $m\times m$ absorbing game. We prove this conjecture, and in fact within a special subclass of absorbing games: for every such $\alpha$, the game realizing it is strictly absorbing and recursive, i.e., every action pair is absorbing with positive probability and all non-absorbing stage payoffs are zero; when $\alpha>0$ it can moreover be taken positive recursive, with positive absorbing payoffs. As a corollary, the set of undiscounted values of rational $m\times m$ absorbing games is exactly the set of real algebraic numbers of degree at most $m$.

In derivative-free optimization (DFO), one minimizes functions for which the gradient is unavailable or expensive to compute. In many applications, objective function values and gradients are noisy due to simulations or system randomness. A class of standard direct-search methods for DFO accept a trial point when it decreases the objective function by an amount proportional to the squared stepsize. However, when applied to complex landscapes, such a requirement may trap the algorithm in a neighborhood of sub-optimal solutions. We study a non-monotone direct-search alternative where the trial function value is compared with the largest objective function obtained through the $M$ most recent distinct iterates. This max-$M$ non-monotone condition permits temporary increases in the objective function and can help navigate narrow curved valleys; however, its theoretical analysis is significantly more challenging due to the lack of monotonic decrease. In this paper, we develop a comprehensive complexity theory for the max-$M$ non-monotone direct-search in both deterministic and stochastic DFO problems. For deterministic objectives, we establish a worst-case iteration bound for a complete poll based on a positive spanning set and an expected iteration bound for a probabilistic-descent poll. We then analyze a stochastic variant using independent function estimates and show the expected iteration complexity under tail-bound assumptions of the stochastic errors. All three results have the standard complexity of $\mathcal{O}(\epsilon^{-2})$, which matches the iteration complexity of monotone direct-search methods. Our theory is enabled by a new family of merit functions that correct the stored objective values by ordered multiples of the squared stepsize, together with a renewal-reward stopping-time argument for the probabilistic methods.

We study why the proximal bundle method (PBM) can perform better in practice when it retains more cutting planes. We consider convex objectives with quadratic growth and an unknown piecewise-smooth structure. Our key observation is that retaining sufficiently many cutting planes allows PBM to exploit the objective's piecewise-smooth structure and behave as if it were optimizing a smooth function. We provide a theoretical explanation for the observed linear convergence of PBM on piecewise-smooth objectives when it retains sufficiently many cutting planes.

Energy-aware scheduling has become a central challenge in modern manufacturing environments. As industries increasingly aim to reduce operational costs and carbon emissions, aligning production activities with electricity tariffs is increasingly important. The integration of battery energy storage system (BESS) further enhances the potential to reduce energy purchase costs; however, incorporating such systems introduces complex interdependencies between job scheduling and BESS management decisions, significantly complicating the problem structure.\\In this work, we study the problem of minimizing the energy cost of executing a set of jobs on a single machine within a fixed time horizon, where electricity prices follow a Time-of-Use (TOU) tariff. In addition, we consider the presence of a BESS that can be charged from the grid and discharged during high-price periods to reduce overall energy costs. To efficiently address this problem, we propose and analyze two matheuristic algorithm variants designed to effectively coordinate production scheduling and BESS usage decisions.

We study asynchronous continuation replanning in a two-population linear–quadratic mean field game with deterministic open-loop controls and heterogeneous, possibly erroneous initial information. At exogenous opportunities, one population recomputes its continuation response from the physical aggregate state reached, with the opponent's active plan frozen. The state–plan target needed to initialize replanning is recoverable from local aggregate observations if and only if a kernel-inclusion condition holds; full recovery of the hidden initial input is unnecessary. For revisions accumulating before the terminal time, spectral stability of the limiting two-response cycle yields bounded physical execution and a finite state left limit. The zero-extended continuation plans then converge strongly to the mean field equilibrium restarted from that state. A scalar counterexample shows that well-posed continuation problems and a convergent physical state can coexist with divergent remaining plans. We also construct an exact causal implementation from local aggregate observations and establish finite-population robustness on every fixed finite opportunity prefix. Revision records match with probability tending to one, and the mean-square implementation error truncated to the matching event is \(O(N_*^{-1})\), where \(N_*\) is the smaller population size.

We stabilize stationary states of coupled multi-species McKean–Vlasov equations by feedback control, extending a spectral approach based on the Wasserstein Hessian and a Riccati equation from a single density to a product of Wasserstein spaces. We give necessary and sufficient conditions for local exponential stabilization at a prescribed rate when only one species is actuated, together with the minimum number of control inputs required. For two species, the condition reduces to a rank condition on the Fourier coefficients of the cross-interaction kernel over the eigenspaces of the unactuated species' linearized dynamics. A two-community noisy Kuramoto model and a cell-sorting model illustrate the results.

We derive the $\Gamma$-limit of convolution-type and discrete energies subject to Dirichlet boundary conditions on periodically perforated domains at the critical exponent. We assume that the length-scale of the non-local interactions is much smaller than the side-length of the cubic perforations and prove that a separation of scales occurs. Exploiting the analogies between the variational frameworks of our interest, we employ a unified argument to overcome the technical difficulties that arise from the scaling invariance of the energies. Our multiscale analysis yields a novel observation that is not related to the non-local nature of the functionals, but rather to the analysis at the critical exponent: we prove that the energy density of the $\textit{strange term}$ is convex, even in the vector-valued setting.

This paper develops an explicit spectral-contraction approach to studying fully coupled nonlinear forward–backward stochastic difference equations on infinite horizon and their applications to stochastic control. The estimates for this fully coupled system are assembled into an explicit two-dimensional nonnegative matrix. Its spectral radius yields an explicit sufficient discount threshold, and a corresponding equivalent weighted product norm is constructed to establish the contraction property. Moreover, for an infinite-horizon control problem with convex control constraints and an accumulated discounted running cost, we derive a Pontryagin-type stochastic maximum principle, its equivalent pointwise normal-cone formulation, and a verification theorem. Finally, a recursive risk-adjusted portfolio example is given to demonstrate the applications of our theoretical results, and a projection-type sufficient optimality condition for this example is derived.

In this paper, we consider the classical time-optimal elevator problem in the presence of higher-order state constraints. The main contribution is a constructive indirect solution framework based on a Pontryagin Maximum Principle specifically formulated for higher-order constrained systems. First, we derive specialized optimality conditions and analyze the resulting structure of extremal trajectories. Second, we show that the infinite-dimensional optimal control problem can be transformed into a finite-dimensional system of nonlinear algebraic equations involving switching times, boundary-contact times, and multiplier parameters. This provides a computationally efficient procedure for calculating candidate optimal trajectories using standard nonlinear equation solvers. Third, we characterize the geometry of boundary contacts and explain the emergence of contact-chattering phenomena in higher-order constrained systems. In particular, we show that normal extremals cannot evolve along the constraint boundary for a positive amount of time, although sequences of boundary contacts may accumulate indefinitely. The proposed approach is demonstrated on a fourth-order elevator model, and the computed solutions are independently verified using a direct IPOPT-based optimization method.

Seasonal reservoir operation requires optimization methods that remain reliable under strongly periodic and uncertain inflow dynamics. We present InFlow, a verification-oriented framework combining periodic stochastic dual dynamic programming (SDDP), entropic robustness, control-space Bellman annealing, and an independently discretized Hamilton-Jacobi-Bellman (HJB) benchmark. The entropic transition operator is interpreted as a KL-penalized worst-case expectation and linked to fixed-radius KL ambiguity sets through Lagrange duality. We establish a uniform soft-Bellman approximation bound and prove the validity of Gibbs-tilted SDDP cuts. The benchmark relies on a normalized seasonal Cox-Ingersoll-Ross inflow process whose positivity condition fails during the driest weeks, requiring explicit treatment of the degenerate boundary. Independent verification shows strong agreement between HJB storage gradients and HiGHS storage-balance duals, with water-value correlations of 0.995 and 0.996 for the standard and exact-cut-only implementations. Refining the HJB grid from 61^2 to 81^2 states changes the annual mean water value by only 0.6% and the peak value by less than 0.1%. Out-of-sample simulations based on 3,000 independent trajectories show that a robustness level of gamma=5 increases nominal mean cost by 4.6% while reducing CVaR90 by 11.2%. Under a dry-season stress scenario with inflows reduced by 40%, the mean-cost difference becomes statistically insignificant whereas CVaR90 improves by 11.5%. A robustness sweep confirms monotonicity of the risk-adjusted value. Finally, a two-reservoir cascade reproduces grid-oracle marginal water values with correlation 0.999 and recovers the predicted upstream-to-downstream value ratio of two. InFlow is intended as a transparent methodological benchmark rather than a calibrated operational reservoir model.

The dynamic mode decomposition (DMD), along with its variant for nonlinear systems, the extended DMD (EDMD), are powerful tools for the extraction of meaningful spatio-temporal characteristics of (non)linear dynamical systems from measurement data. Despite some efforts to handle the identification task when dealing with real-world data, the decomposition based methods face a critical challenge: real-world data has two inherent sources of uncertainty, the process and measurement noise. Subspace identification methods, are robust tools able to provide accurate state-space models for multi-variable linear systems directly from input-output data. Combining these two methods, we introduce the p-q quasi-norm Subspace Extended Dynamic Mode Decomposition (pqSEDMD). An algorithm that uses our previous improvements to the EDMD by the use of a p-q-quasi-norm reduction on an orthogonal polynomial basis, the pqEDMD algorithm, along with subspace identification methods. The result is a robust approximation of nonlinear systems in a linear function space, combining the strengths of the two methodologies. Throughout the paper we will use the Duffing oscillator as a benchmark problem to show the effectiveness of the algorithm and illustrate many important aspects related to the development.

We study early stopping for a continuous-time model of stochastic gradient descent (SGD) in ill-posed linear inverse problems. We consider an a posteriori stopping rule based on the discrepancy principle, which stops the dynamics once the residual reaches the noise level. Unlike deterministic gradient flow, the stochastic dynamics exhibits persistent multiplicative fluctuations whose amplitude scales with the residual itself. Under a source condition on the initial error and a polynomial bound on the spectrum of the empirical covariance operator, we prove that, with high probability, the error at this random stopping time achieves the minimax rate over the corresponding source class, up to a logarithmic factor. Our results therefore establish the discrepancy principle as an adaptive regularization strategy for continuous-time SGD that is minimax-optimal up to a logarithmic factor, despite the persistent fluctuations induced by stochastic sampling. To our knowledge, this is the first convergence-rate result for a continuous-time model of stochastic gradient descent in inverse problems under an a posteriori stopping rule: existing analyses, including those of variance-reduced variants, bound the stopping time but do not quantify the error attained there.

Sequential extremality and stationarity properties are discussed with the emphasis on those corresponding to fixed sequences of translations. Exact quantitative characterisations of the properties are provided. We show, in particular, that the sequential (as well as conventional) extremality and approximate stationarity properties possess certain stability, while the (non-approximate) stationarity does not. Dual necessary conditions for the sequential extremality and stationarity properties with fixed and non-fixed sequences of translations are established. A version of the sequential extended extremal principle is formulated. In the statements, we employ certain generalised separation conditions $(GS)$ and $(GS_\alpha)$ as well as a complementary primal-dual condition $(PD)$. To illustrate the model, we prove dual optimality/stationarity conditions for a constrained minimisation problem in which the minimal value is not necessarily attained.

Optimal scheduling of combined chemo-immunotherapy is often formulated as a control-affine optimal control problem, which generically yields boundary-selected (bang-bang-type) protocols unless singular arcs occur. This structure, while convenient, neglects saturating pharmacodynamics at high dose rates. We incorporate Michaelis-Menten saturation directly into the therapy channels, making the dynamics non-control-affine and the Hamiltonian nonlinear in each control. With strictly positive exposure penalties, the Hamiltonian admits an explicit three-regime pointwise minimization law: each input is chosen at the lower bound, the upper bound, or as a unique interior minimizer. On interior intervals the strict Legendre condition holds, so continuously modulated dosing arises as a regular interior extremal rather than a singular-arc or smoothing artifact. The resulting transcription is nonconvex; we therefore use a hybrid Quantum Particle Swarm Optimization (QPSO)-Sequential Quadratic Programming (SQP) pipeline, where QPSO provides a constraint-aware warm start and SQP enforces feasibility and Karush-Kuhn-Tucker (KKT) optimality on a collocation grid. Costate reconstruction corroborates the Pontryagin Minimum Principle (PMP) structure and the predicted boundary/interior regime transitions.

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost, respectively, for any $\beta \in [0,1)$, over a time horizon of $T$. We also show that any weakly adaptive online algorithm with $O(T^\beta)$ regret has a movement cost of $\Omega\left(T^{\frac{1-\beta}{2}}\right)$ for any $\beta \in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear anytime regret has a movement cost of $\Omega\left(\log T\right)$.

Optimal control in cancer chemotherapy is challenged by tumor heterogeneity and mutations, which complicate treatment effectiveness. Traditional methods, such as Pontryagin's maximum principle (PMP), are often hindered by their reliance on an initial guess for the costate equation, affecting accuracy and convergence. To address these limitations, this work introduces a hybrid quantum particle swarm optimization (QPSO) method based on regularization. QPSO is employed for global exploration to approximate the optimal control trajectory, followed by a regularization-based refinement to ensure smoothness and consistency with optimality conditions. The Hamiltonian function is used for first- and second-order optimality checks, verifying solution quality. Numerical case studies explore various drug effectiveness functions, demonstrating the role of periodic and localized drug delivery in achieving robust tumor suppression and providing insights into the impact of different drug combinations on optimal chemotherapy strategies.

Bilevel optimization has recently attracted growing attention, particularly in the development of efficient numerical methods. Despite substantial progress on optimistic bilevel optimization, pessimistic bilevel optimization (PBO) remains much less explored, especially the design of fully first-order, single-loop gradient-based methods. To address this gap, we propose a smooth approximation of PBO through reformulation, penalization and regularization, and establish convergence guarantees in terms of both minimizers and stationarity. Building on this framework, we then develop two single-loop algorithms for deterministic and stochastic PBOs, respectively. Both use only first-order gradient information and avoid second-order derivatives and inner-loop subproblem solves. Non-asymptotic convergence rates for the proposed algorithms are established to provide theoretical guarantees. Through a systematic empirical study of both synthetic and practical problem instances, we demonstrate that our algorithms are highly effective and efficient. In particular, our results on spam classification and Smart Predict-then-Optimize further illustrate PBO's advantages over its classical optimistic bilevel counterpart, highlighting its strong potential for practical modeling and the delivery of robust solutions.