Papers

This work proposes a biomechanical registration framework designed to combine functional optical-mapping data with structural high-resolution MRI of cardiac tissue. The two acquisitions are performed in different mechanical configurations: optical mapping is obtained after stretching the sample, whereas MRI is acquired in a fixed resting state. To estimate the deformation between these configurations, the tissue is modeled as a nearly incompressible transversely isotropic hyperelastic material, and the traction amplitude is identified from optical-map marker positions through an inverse problem. Two-and threedimensional simulations show physically admissible deformations and provide a first proof of concept for the proposed strategy.

Retuning feedback controllers in critical processes is challenging when experiments are costly, open-loop excitation is infeasible, and candidate controllers must be screened before deployment. The existing Adaptive Sampling for Counterfactual Explanations (ACE) method provides a sample-efficient approach for finding minimal controller-parameter changes that recover prescribed closed-loop performance, but it does not explicitly account for stability of candidate controllers proposed during the search, which may be destabilizing under plant uncertainty. This paper introduces UQ-ACE, a stability-aware retuning method based on Bayesian SINDy. In UQ-ACE, closed-loop trajectories are used to infer a posterior distribution over local dynamical models, from which candidate controllers are screened for stability before implementation. Extensive simulations show that UQ-ACE achieves valid retunings with few closed-loop experiments while preventing deployment of unstable candidates.

We propose a smoothed augmented Lagrangian method (ALM) for solving mathematical programs with vertical complementarity constraints (VCCs). The augmented Lagrangian function involving the VCCs is smoothed by the Lasry–Lions double envelope, which is an extension of the Moreau envelope that enjoys enhanced smoothness properties. This construction yields a smooth unconstrained subproblem within the ALM framework that can be efficiently solved by many fast optimization algorithms. In the limiting setting, we establish global convergence to Mordukhovich- and Clarke-stationary points. For the convergence rate, we show that, under appropriate assumptions, the smoothed ALM achieves Q-linear convergence. We further provide a worst-case iteration complexity analysis for computing an approximate stationary point of the original problem or of an associated feasibility problem. Preliminary numerical experiments demonstrate the effectiveness of the proposed method.

Coordinate descent solvers for regularized generalized linear models (GLMs) differ in how they update the intercept. This seemingly routine choice can substantially affect convergence when the response is imbalanced. For direct coordinate descent on the original GLM loss, we compare a conservative step based on worst-case curvature, a safeguarded Newton step based on current curvature, and repeated Newton steps that solve the intercept subproblem. The conservative update can leave much of the interaction between the intercept and the coefficients unresolved, producing a slowdown when that interaction controls convergence. A single Newton step avoids this attenuation and, in our experiments, follows nearly the same outer trajectory as exact minimization while requiring fewer inner steps. Simulated and real-data experiments, including comparisons across six production solvers and a controlled intervention within skglm, support this explanation. We also show how the same mechanism appears in quadratic-approximation methods and extends to multinomial models with rare classes. We recommend a safeguarded Newton update for direct coordinate descent and full intercept optimization within frozen local quadratic surrogates.

We study a class of congestion games with a continuum of heterogeneous players and a finite set of resources, without assuming the existence of a potential function. Players differ in their individual characteristics and may contribute differently to congestion, which is represented through general aggregate features of the population choosing each resource. We characterize the conditions under which such a game admits a potential and show that this structure generally fails when players have heterogeneous impacts on congestion. To analyze the non-potential case, we introduce the concept of Stochastic Variational Nash Equilibrium, in which players observe idiosyncratic cost shocks before selecting a resource. We show that equilibrium choices are pure conditional on players' characteristics and shock realizations, whereas the induced aggregate allocation is probabilistic. Under Gumbel-distributed shocks, the equilibrium is equivalent to a continuum-player logit Quantal Response Equilibrium. Using a fixed-point formulation and the Schauder–Tychonoff theorem, we establish existence for every positive noise level and recover a pure Nash equilibrium in the zero-noise limit. Under a coordinate-wise monotonicity condition on aggregate interaction costs, we further prove uniqueness and convergence of the regularized allocations to the pure equilibrium. An application to electric vehicle charging illustrates how heterogeneous energy demands jointly affect station choice, congestion, and market shares.

This paper studies the spectral compressed sensing problem, which aims to recover an $r$-spectrally-sparse signal of length $N$ from linear compressive measurements. We reformulate the problem as the recovery of a low-rank Hankel matrix. Motivated by this formulation, we propose a Lifted Restricted Isometry Property (LRIP) for the measurement matrix and show that Gaussian measurement matrices satisfy LRIP under certain conditions. Unlike classical RIP, LRIP imposes a rank constraint in the lifted matrix domain while taking measurements in the signal domain through the adjoint Hankel operator. To solve the resulting nonconvex problem efficiently, we introduce a Fast Iterative Hard Thresholding (FIHT) algorithm. Based on LRIP, we show that FIHT converges linearly to the true signal with high probability using $\mathcal{O}(r \ln r \ln^2 N)$ scaled Gaussian measurements, which is near-optimal up to log factors and improves the state-of-the-art order of $r^2$ for nonconvex Hankel methods. Extensive numerical results demonstrate the superiority of FIHT as compared to state-of-the-art convex methods in terms of phase transitions, computational time, and robustness to noise.

We study two inclusion-wise extremal variants of the 0-1 Knapsack Problem (KP). Given $n$ items with integer weights and nonnegative costs or profits, the Minimum-Cost Maximal Knapsack Packing Problem seeks a minimum-cost packing for capacity $c$ to which no item can be added, while the Maximum-Profit Minimal Knapsack Cover Problem seeks a maximum-profit cover for threshold $d$ from which no item can be removed. Both problems are weakly $\mathrm{NP}$-hard and pseudo-polynomially solvable, as is the classical KP, but their approximability is fundamentally different: unless $\mathrm{P}=\mathrm{NP}$, with positive values neither problem admits a polynomial-time $2^{p(n)}$-approximation for any polynomial $p$; allowing zero values rules out every multiplicative factor. Even with unit costs for packing and unit profits for covering, neither admits an $n^{1-\varepsilon}$-approximation for any fixed $\varepsilon>0$, whereas a trivial $n$-approximation exists. We identify restrictions that restore approximability. If every item weighs at least a fixed fraction of the capacity, respectively of the threshold, both problems can be solved exactly in polynomial time. For the cover problem, if each weight $w_i$ satisfies $w_i-1\ge\delta(d-1)$, we give an approximation scheme for arbitrary profits whose running time is polynomial in $n$, $1/\varepsilon$, and $1/\delta$. If values equal weights, a fully polynomial-time approximation scheme (FPTAS) was already known for the packing problem; we give an FPTAS for the cover problem, improving on the previously known factor $2$. If value-to-weight ratios differ by at most a factor $\kappa$, we obtain a $(1+\varepsilon)\kappa$-approximation for both problems.

In this article, we study optimal control problems driven by mean-field forward-backward stochastic differential equations (FBSDEs). The aim is two-fold: (i) to provide the unique optimal control; (ii) to establish classical solutions of the associated generalized Hamilton-Jacobi-Bellman (HJB) equations on the space of probability measures. For (i), we study the coupled system arising from the maximum principle, which involves both the state FBSDEs and the adjoint FBSDEs. This system is non-standard and more complicated than that in standard mean-field type control (MFTC) problems. By extending the method of continuation in the coefficients to our framework, we establish the well-posedness result, and subsequently obtain the differentiability of the value functional with respect to the distribution argument — thereby extending existing results for MFTC problems. For (ii), since the flow property does not generally hold for control problems of FBSDEs, standard dynamic-programming arguments cannot be applied directly. We therefore develop probabilistic approaches to the classical solvability of generalized HJB equations under three distinct sets of conditions. The auxiliary processes arising in these arguments differ significantly from those in standard MFTC problems. We establish new connections between the value function and the adjoint processes, derive equations for the relevant decoupling field, and obtain first- and second-order generalized HJB equations, including a coupled system on the space of probability measures. The standard HJB equations for MFTC problems are recovered as special cases. Our approach yields classical solutions without requiring non-degeneracy of the diffusion.

This paper shows how a Schrodinger bridge (SB) structure emerges from maximum-entropy (MaxEnt) distribution steering of discrete-time nonlinear systems. We also establish that the resulting bridge can be realized by an optimal control policy. We consider general state and input costs, non-Gaussian endpoint distributions, and input constraints. Assuming the existence of a feasible policy with finite objective value, we show that the state-path measure under a MaxEnt optimal control policy is the unique SB. The reference measure of this SB is induced by an optimal policy of the corresponding terminal-unconstrained problem. Moreover, we give an equivalent characterization of the optimal state-path measure using a killed reference and its normalized Feynman–Kac measure, thereby interpreting the state cost as killing. We then characterize the finite-cost feasibility by the existence of an endpoint coupling having finite Kullback–Leibler divergence from the reference endpoint distribution. For compactly supported marginals with bounded target density, we derive reachability-based sufficient conditions guaranteeing this feasibility. We further provide specializations to fully actuated control-affine and reachable linear systems.

In stochastic control problems involving optimal portfolio choice, one often seeks an optimizer of some local criterion (dynamic programming value function) generated by some global criterion (e.g., expected utility). The local optimizer then typically acts as a stochastic integrand with respect to some multivariate stochastic process (e.g., asset prices). This paper takes a closer look at local criteria in the form of “local expected utility”. Since the existence of the local optimizer can be established under very mild conditions, it is crucial to understand under what circumstances the local optimizer has good integrability properties. It is shown that the reasonable asymptotic elasticity condition of Kramkov and Schachermayer (1999) applied to the local utility function plays a key role in this context.

Demand for wireless communication is growing, and commercial satellite services are expanding with it, increasing pressure on a finite radio spectrum. Allocation efficiency directly limits how many users a satellite can serve. We study how to assign frequency sub-channels to the carriers of a multi-beam satellite, maximising the total priority of admitted carriers while keeping each within its interference tolerance. Earlier work on this interference-limited channel allocation problem placed each carrier in a single continuous frequency block. Removing this restriction can admit more carriers, but a direct integer programming model cannot solve the largest test problems in practical time, so new methods are needed. We prove the problem computationally hard, reformulate it around how carriers share sub-channels, and derive three bounds on the optimum. We develop heuristics, including matheuristics combining local search with optimisation models, and an orchestrated portfolio that schedules them, recombines their solutions and stops early once its result is proven optimal or no longer improves. On 24 problems with up to 500 carriers, the best heuristics come within about 1% of the best known solution on average. The portfolio matches or exceeds the best heuristic for each problem on 23 of 24 with equal weights and 22 with priority weights, and is up to about 5% better on larger, more congested problems. It requires about a quarter of the time to run every heuristic and keep the best, and under half that of its heuristics run in a fixed order, which return slightly worse solutions on average.

This paper develops theoretical and numerical extensions of Birkhoff pseudospectral transcription for aerospace optimal control problems. Building on the recently proposed body of knowledge, a Birkhoff integration operator is derived for flipped Legendre Gauss Radau (fLGR) grids, drawing connections to regular LGR grids and to the favorable conditioning properties that make Birkhoff interpolants particularly appealing. We also propose the enforcement of differential constraints via pseudospectral operators, and the primal and dual equivalence of this approach to state augmentation is established for control-rate limits. The numerical developments include the formulation and solution of heterogeneous multiphase optimal control problems, parameterized non-affine time mappings, composition of affine scaling and equilibration methods, and a hybrid derivative strategy to combine linear pseudospectral operators with numerical derivatives for nonlinear functions. These methods are implemented in the RAPTOR (Rapid Algorithm for Pseudospectral Trajectory Optimization and Robustification) transcription framework and assessed through feature-oriented benchmarks: a one-dimensional landing problem, a planar orbit-raising scenario, an atmospheric entry problem, and a multiphase reusable launch vehicle problem with simultaneous booster-recovery and upper-stage-insertion trajectories.

This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices $D$ and $D+\bar D$ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value $P(0)$ of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.

In answering the main criticism of the classical robust optimization that the size of the uncertainty set has to be determined beforehand, a recently proposed Joint Estimation and Robustness Optimization framework treats the size of the uncertainty set as a variable and maximizes it subject to some robust conditions over the uncertainty set. This paper studies its effect on portfolio selection when both the robust return and risk constraints are built into Joint Estimation and Robustness Optimization framework. Although this framework does not directly pursuit any optimality of portfolio return or risk, we report a surprising result that Joint Estimation and Robustness Optimization framework actually selects efficient portfolios. This is proved by reformulating the robust risk constraint as a tractable form through the S-procedure. Moreover, a closed-form formula was derived when there is only the budget constraint. An efficient algorithm is proposed when more constraints are added. Numerical experiments illustrate the validity of the reported results. This research paves the way for Joint Estimation and Robustness Optimization framework to be applied to the portfolio selection problem with return and risk related robust constraints.

GPU-based first-order methods are emerging as an approach for solving large-scale linear programming (LP) problems. We present H-PDLP, a GPU first-order LP solver based on a generalized primal-dual hybrid gradient (PDHG) method for a homogeneous self-dual embedding of the LP, and study its numerical performance. The solver applies the restarted reflected Halpern iteration to the generalized PDHG operator, with the embedding encoding both primal-dual optimal solutions and certificates of primal or dual infeasibility. We derive a compact embedding for LPs with two-sided constraint and variable bounds in which the bounds enter through box cones, and we show that the resulting box-cone projections reduce to one-dimensional root-finding problems, so that each iteration remains matrix-free and well suited to GPUs. On the theoretical side, we establish conditions under which the iterates remain uniformly bounded away from the trivial zero solution of the embedding and derive complexity guarantees for the method. We also introduce MIPLIBInfeas, a reproducible benchmark of 229 infeasible and 105 unbounded LPs derived from MIPLIB 2017. Numerical experiments demonstrate the effectiveness of the proposed approach on both feasible LPs and infeasible or unbounded LPs. The source code for H-PDLP is publicly available at https://github.com/MIT-Lu-Lab/H-PDLP.

We study a broad class of stochastic transport problems under controlled McKean-Vlasov dynamics, allowing control of both drift and volatility, with weak terminal constraints. In contrast to the classical Schrödinger problem, the reference dynamics depend on the evolving law of the process, leading to nonlinear optimization problems on path space, while the terminal constraint is formulated through weak optimal transport, permitting a flexible treatment of distributional targets. This framework encompasses several problems of current interest, including the mean field Schrödinger problem, martingale optimal transport, and Schrödinger-Bass bridges. We establish three main results. First, we prove the existence of optimal controls under general coercivity, continuity, and convexity assumptions. Second, we derive a convex dual representation that extends weak Kantorovich duality and stochastic optimal transport duality to controlled McKean-Vlasov dynamics. Finally, for the mean field Schrödinger problem, we prove convergence, at the level of value functions, of the associated finite-particle Schrödinger bridge problems to the mean field limit, thereby identifying the nonlinear problem as the exact infinite-particle limit of finite-particle Schrödinger bridges. As a byproduct, we establish the existence of an optimal mean field Schrödinger potential, providing a variational foundation for duality in nonlinear stochastic transport.

We develop analytical sufficient conditions for certifying the local asymptotic stability of nonlinear autonomous systems based on Lyapunov functions derived from their finite-order Taylor polynomial approximations. This approach translates the stability analysis problem from the original nonlinear dynamics to a more computationally tractable polynomial domain. The derived conditions allow for the explicit calculation of a guaranteed inner approximation of the region of attraction ($\mathrm{ROA}$). We show how sum-of-squares (SOS) programming and grid search can be used to systematically search for polynomial Lyapunov functions that yield a large certified threshold radius and extract a certified invariant set inner approximation of the $\mathrm{ROA}$ of the nonpolynomial original system. The framework is then applied to a control tuning problem: finding reduced-effort Linear Quadratic Regulator (LQR) controllers that ensure that a given initial state of interest belongs to the certified $\mathrm{ROA}$ of the nonpolynomial closed loop system. The effectiveness of the overall methodology is demonstrated using a pendulum with nonlinear damping and a nonlinear control system.

Inference-time alignment steers diffusion models toward downstream objectives without costly fine-tuning, but existing methods typically optimize expected cost (or reward) without explicitly controlling tail risk. In this work, we propose a risk-averse framework for inference-time alignment of diffusion models that balances expected cost and Conditional Value-at-Risk (CVaR) under trajectory-level Kullback-Leibler regularization. We characterize the optimal path distribution and reverse transitions as functions of the CVaR threshold, reducing the alignment problem to a one-dimensional optimization problem over the threshold. We show that this one-dimensional optimization problem can be nonconvex in general, establish sufficient conditions for convexity, and introduce quadratic regularization to enable efficient derivative-free ternary search. Under suitable assumptions, we provide finite-sample guarantees for threshold optimization and value estimation, and combine them into a terminal total-variation bound relative to the regularized target under exact resampling. Experiments on synthetic data and image generation using Stable Diffusion demonstrate improved tail performance over risk-neutral baselines while maintaining competitive average cost.

In this paper, we study the active vibration control of an elastic beam using reinforcement learning (RL). The physical system is modeled by a modified Timoshenko-Ehrenfest type linear partial differential equation with simply supported boundary conditions. The dynamics are approximated by a reduced-order model (ROM) obtained through eigenvalue expansion and truncated Galerkin projection. Control is applied through boundary actuation. The resulting finite-dimensional system is formulated as a continuous-time Markov decision process and solved numerically using the Proximal Policy Optimization (PPO) algorithm. All modeling steps, from physical formulation to temporal discretization and control design, are explicitly described. A systematic comparison is performed between the controller learned via PPO and the classical linear quadratic (LQR) regulator. The analysis highlights the strengths and limitations of each approach, both from a theoretical and practical point of view, with particular emphasis on energy dissipation mechanisms and actuator constraints.

Safety verification of a Markov decision process (MDP) asks whether the probabilities of reaching several failure states jointly lie in a convex safe region under every scheduler. Watanabe and Unno (2026) prove safety by refining an outer approximation of the failure probabilities, where each query bounds a nonnegative weighted sum of the probabilities. Their rule selects weights by the largest Euclidean violation. We present a unified framework to measure violation in an explicitly chosen error unit: directions are scored by the support-function gap between the outer approximation and the safe region, normalized by the support function of the unit. We show that the maximal score, when positive, equals the smallest enlargement of the safe region in multiples of that unit that contains the outer approximation. The Euclidean unit recovers the earlier rule; two new rules are generated by the unit cube and the safe region itself. We experimentally evaluate how the choice of measurement rule affects the performance of safety verification for MDPs.

We study the stability of the variable coefficient wave equation $u_{tt}-u_{xx}-(1+x^{2})u_{yy}+\chi_{\mathbb G}u_{t}=0$ on the rectangle $(-1,1)\times(0,\pi)$ with Dirichlet boundary conditions, where the viscous damping acts on a vertical strip $\mathbb G=(\eta-\delta,\eta+\delta)\times(0,\pi)$. We establish a precise relationship between the location of the damping region and the rate of energy decay. We prove that the associated semigroup is exponentially stable whenever the closure of the damping strip meets the line $x=0$, i.e., $|\eta|\leq\delta$, despite the existence of rays that never enter the damped region. When the damping region is strictly bounded away from $x=0$, the semigroup decays at the sharp rate $(\log(2+t))^{-1}$. The proofs combine separation of variables, one-dimensional resolvent estimates, a reflection identity for the harmonic oscillator, and Gaussian quasimodes. We then consider the spectral Galerkin approximation based on the Dirichlet eigenfunctions of the constant coefficient Laplacian and show that, although the sequence of semigroups resulting from the approximation converges to the continuous one, it fails to be uniformly exponentially stable.

We maximize the weight of edges covered by selected vertices, together with nonnegative individual rewards, under an arbitrary matroid constraint. The continuous coverage objective is quadratic and need not be concave. Our method searches projections onto a small set of directions and solves a concave subproblem for each projection center. A coverage-specific inequality converts the subproblem bounds into a multiplicative guarantee, while matroid rounding restores an integral solution. Let $A$ be the weighted adjacency matrix and $D$ the diagonal matrix of singleton values, restricted to positive-value elements after objective-preserving preprocessing. Set $\varrho=\lambda_{\max}(D^{-1/2}AD^{-1/2})$. For an integer budget $q\ge0$, let $\tau_q$ be the smallest nonnegative threshold for which $A+\tau_qD$ has at most $q$ negative eigenvalues on the span of differences of matroid bases. We obtain the factor $1-\varrho\tau_q/[2(\varrho+\tau_q)]-\varepsilon$ in time $(2+q/\varepsilon)^{O(q+1)}N^{O(1)}$, where $N$ is the binary input length and a zero-over-zero loss is zero. The algorithm returns a rational upper bound on the integral optimum. A separate exact-algebraic construction produces polynomial-bit low-rank rational corrections with positive threshold slack; its proof isolates eigenvalues, keeps eigenspace bases unnormalized, and approximates scalar factors only at the final step. The numerical proposal prototype is not used to justify this complexity statement. Supplied certificates also yield robustness to locally bounded edge additions. These are threshold-dependent coverage guarantees, not a new general spectral-search principle or a universal approximation improvement.

Selecting vertices covers their incident edges, but a matroid can prevent selecting the combinations that cover the most weight. We study whether pair-selection information improves rounding from a semidefinite relaxation of this problem. A feasible point $(x,Y)$ records individual and paired selection values. Its relaxation objective is $U$, and the matrix $Y-xx^\top$ has rank $d$. We compare the feasible marginal vectors obtained by fixing one element as selected or excluded, then round only the best vector. For $d\ge1$, the resulting basis has value at least $(3/4+1/(4d))U$. The procedure is deterministic and polynomial-time for a supplied exact rational point and an independence oracle. Weighting the covariance spectrum by the objective gives a stronger version of the bound. To show the rank factor is sharp, we construct a rank-$2m$ rational linear matroid and $m$ disjoint four-cycles with integral optimum $3m+1$, relaxation optimum $4m$, and an optimal covariance matrix of rank $m$. Thus the relaxation has limiting reciprocal integrality gap $4/3$, despite its conditional constraints. One valid pair-exclusion inequality makes it exact on this family, as an explicit semidefinite certificate shows. The result concerns rounding and the strength of this relaxation; it does not supply a low-rank optimal point or a fixed universal approximation improvement.

Prescribed-time controllers drive the state to the origin at a time $T$ chosen in advance, with gains that grow without bound as $t$ approaches $T$. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with $\|\dot z(t)\|\le L(t)\|z(t)\|$ satisfies $\|z(t)\|\ge\|z(0)\|e^{-\int_0^t L}$. From this: a gain profile $L$ supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over $[0,T)$ is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget $\mathcal B=\int_0^T L$ the best worst-case contraction over the class is exactly $e^{-\mathcal B}$, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes $\|\dot z\|\le L(t)\,\omega(\|z\|)$, first arrival at the origin requires budget at least $\int_0^{\|z_0\|}du/\omega(u)$; arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.

In a smart grid, consumers can also play the role of producers and sellers of energy. Therefore, the control of a smart grid over time becomes a dynamic game with a large number of small players and one or several large players. This is much more complicated than the control of an energy system with a single producer. The framework of mean-field games with common noise allows the transformation of such a complicated model into a control problem for a typical prosumer (producer and consumer in one) optimizing his behavior against a flow of other such prosumers. In our model, we introduce 2 decision variables for such a controller: one describing the amount of electric energy bought from the grid (or sold to it), and the second one describing the charging rate for our battery. The cost function is designed to satisfy two goals: satisfying the demand of the consumer (a single home) and minimizing the payment for energy. Players are allowed to sell/buy energy from the grid. Common noise is hidden in price volatility and sun irradiance, which influence all the players. We compute the optimal controls for our model and simulate solutions on real data. The novelty of our construction, compared to other works on smart grid control, lies in combining the minimization of the cost of energy and sustaining some level of demand satisfaction. We analyze the impact of battery usage on achieving a balance between these two criteria. We also take into account that the smart grid is only part of the energy market, which makes consumer strategies dependent on energy consumption outside the grid.

We investigate a newsvendor problem where a decision-maker observes a $z$-dimensional vector of covariates, and wants to learn how much stock to order given a sample of size $n$ of censored demand data to minimize costs. Demand is censored when insufficient stock is available to meet it, and past stocking quantities are known. We propose a new model-agnostic loss function approach to estimate optimal order quantities using observational data, and characterize the optimal newsvendor policy as the policy that minimizes our proposed loss function. Our method requires only a single estimation step, can be scaled to datasets with hundreds of thousands of data points, and does not suffer from the same instabilities as inverse censoring probability weighting methods. We describe how this loss function can be implemented in linear empirical risk minimization, kernel weights optimization and deep neural network algorithms, and provide excess risk generalization bounds for their out-of-sample performance. Moreover, we show through numerical experiments that our method quickly converges to the optimal policy on synthetic data, and leads to improved performance on real world retail data.

Let $Q \subseteq \mathbb{R}^n$ be a open, convex, and symmetric set and let $f:Q \rightarrow \mathbb{R}$ be a $\vartheta$-self-concordant barrier function, that is also symmetric. We show the associated spectral function $F(X)= \mathcal{O}^2 (f \circ \lambda)(X)$ is a $\mathcal{O}^2\vartheta$-self-concordant function, where $\mathcal{O} = 8\max\{1+3\vartheta, 12\}$, on the domain of all $n \times n$ symmetric matrices $X$, with eigenvalues $\lambda(X) \in Q$. This generalizes the relationship between the standard logarithmic barriers $-\sum_{i=1}^n\log x_i$ and $-\log \det X$ and gives a solution to a conjecture by L.~Tunçel, 1995.

Parametrized gates in a variational quantum eigensolver (VQE) live on a product of unitary groups, which makes Riemannian optimization a natural candidate for improving training. We test this candidate in the clean per-gate setting: each single-qubit gate is optimized directly on U (2) and compared with Adam on an unconstrained Euler-angle chart representing the same physical gate, initialized from the same unitaries and tuned with a separate learning-rate sweep. On a validated hydrogen-chain benchmark, we find no measurable practical advantage from the Riemannian machinery. The direct manifold optimizer and the Euler-coordinate baseline are interleaved in final error and success rate; the classical coordinate baseline is not slower in iterations on the successful cells; and the choices of Riemannian metric, retraction, and momentum vector transport do not change the conclusion. We also identify four failure modes that can manufacture an apparent positive result: a trivial product-state benchmark, a projected-matrix Adam baseline that discards most of its own step, a non-unitary inherited Cayley retraction that lets the reported energy pass below the variational minimum, and a learning-rate grid whose optimum sits at its boundary. Each failure mode produced a plausible multi-seed result in our own workflow before being caught. The practical output of this paper is therefore a controlled null result and a validation checklist for VQE optimizer comparisons.

Fourier-based spectral methods provide compact global approximations, permit simple differentiation and integration, and achieve spectral convergence for smooth periodic functions. Their use in finite-horizon optimal control problems (OCPs) is nevertheless challenged by the generally nonperiodic state and control trajectories: endpoint mismatch induces artificial discontinuities and can destroy spectral convergence. Leveraging Eckhoff-type endpoint-correction techniques, we develop a Fourier-based spectral framework for OCPs with possibly nonlinear dynamics and constraints. The correction improves convergence without enlarging the set of decision variables, yielding an efficient finite-dimensional numerical method. Furthermore, we formulate selected dynamics and obstacle-avoidance constraints through Sobolev residual bounds, which incorporate information about the residuals over the entire time horizon and improve constraint satisfaction. Numerical experiments on benchmark problems demonstrate competitive computational performance, lower residual errors, and improved solution accuracy.

Moment relaxations provide lower bounds for polynomial optimization problems, while the moment matrix carries more information than the bound itself. When the moment matrix satisfies the flat truncation condition, the bound is tight and global minimizers can be extracted from the matrix. In correlatively sparse relaxations, it is common that only some blocks are flat. We study whether such partial flatness still reveals part of a global minimizer. On a unicycle trajectory optimization problem, repeatedly extracting the atoms of flat cliques and fixing them recovers feasible, near-optimal trajectories. We then construct a counterexample whose relaxation is tight and has a rank-one flat clique, yet the unique extracted atom cannot be extended to a global minimizer. Finally, we give sufficient conditions for global extendability that involve only the boundary of a clique. Every extracted atom minimizes its clique objective given its boundary value, and when the bound is tight and the boundary moment block has rank at most one, the extracted atom of the clique is globally extendable.