Papers

Permutation optimization arises whenever the components of a system are fixed but their ordering affects performance. We introduce PRISM, a predictive protocol for permutation optimization that measures a fitness landscape before selecting a search strategy. PRISM uses inexpensive landscape diagnostics, including one-step move autocorrelation and fitness-distance correlation, to predict useful mutation operators, identify when structured search is likely to outperform random sampling, and detect regimes in which search provides little advantage. Across synthetic permutation landscapes, neural architecture benchmarks, scientific machine learning pipelines, and large-language-model instruction ordering, the protocol makes testable predictions about search behavior before optimization begins. Exhaustive instruction-ordering experiments reveal substantial performance variation induced solely by permutation, while cross-model experiments show that useful ordering structure can transfer across model families and task difficulty. Additional experiments demonstrate that instruction ordering remains consequential after prompt wording is optimized, indicating that content optimization and ordering optimization are complementary. The results position PRISM not as a universally superior optimizer, but as a framework for determining when permutation search is useful, which representation and operator should be used, and when simpler alternatives are preferable.

In this paper we consider (MIMO) finite dimensional linear and time-invariant systems and irrational single-input single-output (SISO) transfer functions. In both cases our aim is to describe as best as possible the range of the input-output map $u(\cdot) \mapsto y(\cdot)$. The theory in finite dimension is quite satisfying but rather scarce, the first aim of this paper is to collect the main results in this direction in a comprehensive way. Our second aim is to improve a recent result on the tracking problem for irrational SISO transfer functions, allowing one to completely describe the outputs of such system for inputs $u \in L^2(0,\infty)$. An application is given for a heat equation with polynomial coefficients.

Backward differences are a standard digital realization of derivative action in proportional-integral-derivative control. This note proves that, for multivariable state-space plants, this implementation can be unstable no matter how small the sampling period is. The exact lifted model reveals fast eigenvalues created by the stored previous output sample. Away from boundary spectra, stability is equivalent to ideal loop stability plus Schur stability of the product of the output, input, and derivative gain matrices.

In this paper, we consider an optimal stopping problem with infinite horizon, non-negative pay-offs and non-linear evaluations $\rho_{S,\tau}$ indexed by two indices: $S$ and $\tau$, where $S$ is the time of evaluation and $\tau$ is the time when the pay-off is revealed. The agent's stopping strategies are constrained to be in the set of so-called Bermudan stopping times $\Theta$. Under suitable assumptions on the non-linear evaluations $\rho$ and on the pay-off, we show that a dynamic programming principle holds in this framework. We investigate the existence of $\varepsilon$-optimal stopping times, as well as the existence of optimal stopping times. We show that an $\varepsilon$-optimal stopping time exists. We also prove that the first time when the value family hits the pay-off is optimal if and only if it is finite. We also provide Doob's type convergence for non-negative \emph{$(\Theta, \rho)$}-supermartingales in the case where $\rho_{S,\tau}=\rho_S$ depends on the first index only. We provide an example from BSDEs with infinite horizon.

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $\epsilon$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If \(C_{\mathrm H}(T,L)\) denotes the optimal \(L^2\) null-control cost for initial data in \(H^{-1}(0,L)\), we prove that \[ C_{\mathrm H}(T,L) = \exp\left(\frac{\kappa_*L^2+o(1)}{T}\right), \qquad \kappa_* = \frac{\Gamma(\frac14)^4}{8\pi^3} \simeq 0.696601964842838, \qquad T\to0^+. \] The constant \(\kappa_*\) coincides with the upper-bound constant obtained by Dardé and Ervedoza (2019, ANPDE), which was expressed there through a convergent series. This closes the gap between the lower bound obtained in by Lissy (2015, JDE) and the upper bound established by Dardé and Ervedoza.

We study feedback stabilization for the discrete-time system $x_{t+1}=f(x_t)+u_t+w_{t+1}$ in $\mathbb{R}^d$ with unknown $f$ and arbitrary bounded disturbances. For scalar plants, the sharp feedback capability threshold under generalized Lipschitz uncertainty is $3/2+\sqrt{2}$. We treat fully coupled vector-valued systems, where scalar order and interval recursion are unavailable and coupling precludes a coordinatewise reduction. We introduce a response-history escape game in which the adversary seeks a finite envelope and an unbounded state radius. Borel determinacy ensures that exactly one player has a winning strategy at each slope. We prove that the same player wins from every finite response history, and slope monotonicity gives an independently defined game value $\Gamma_d$. We prove that $\Gamma_d$ is finite and is the strict feedback capability threshold for the plant problem. If $L<\Gamma_d$, one causal feedback law stabilizes every plant in the uncertainty class against every bounded disturbance sequence. If $L>\Gamma_d$, for every causal feedback law there exist a plant in the same class and a bounded disturbance sequence such that the closed-loop state sequence is unbounded. The proof uses one controller for all subcritical slopes and a realization in a Hilbert space based on the Kirszbraun–Valentine extension theorem. An explicit nearest-neighbor law gives a lower bound above one in every finite dimension, including $\Gamma_2\ge 2/\sqrt{3}$. Dimension monotonicity gives $\Gamma_d\le\Gamma_1$, and comparison with the scalar theory yields $\Gamma_1=3/2+\sqrt{2}$.

In a finite recursive game in the sense of Everett, both players have stationary epsilon-optimal strategies for every epsilon>0. Frederiksen and Miltersen strengthened this result by showing that the strategies for all sufficiently small epsilon can be encoded by finitely many monomials: at every state, all but possibly one of the action probabilities are constants times integer powers of epsilon. The resulting finite symbolic object specifies a strategy for every sufficiently small accuracy. Their proof uses semialgebraic selection and Puiseux series. We give an alternative elementary proof of this regularity theorem for recursive games. We start with stationary strategies that guarantee vectors approaching the value through Everett's one-sided region. After fixing their support, we express, for each pure stationary reply, all absorption probabilities as quotients of directed-forest polynomials with nonnegative coefficients and a common positive denominator. Each payoff is a fixed signed linear combination of these quotients. We then compress the asymptotic orders of the finitely many forest monomials into one integer weight vector. This proof uses neither semialgebraic selection nor Puiseux series. Furthermore, for rational games with a fixed number N of active states, we present a deterministic polynomial-time algorithm that computes a monomial family exactly. It returns all algebraic coefficients in one ordered real univariate representation. The representation length and running time are at most L^{(N+1)^{O(N)}}, where L is the input length.

A convex polygon is called small if its diameter is at most one. Reinhardt proved the universal perimeter bound $\operatorname{perim}(P)\le U_n:=2n\sin(\pi/(2n))$, and the bound is attained whenever $n$ has a nontrivial odd divisor. The remaining power-of-two cases have resisted exact solution beyond $n=8$. We give computer-assisted proofs of the first three cases, $n=16,32,64$, and in each case prove uniqueness of the maximizing congruence class. The proof architecture is common to all three cases: pass to the difference body $P-P$; encode its reconstruction by a sign code; prove that every global maximizer is saturated, so all difference-body vertices lie on the unit circle; localize every competitive configuration near the regular angle vector; exhaustively screen the sign codes using exact arithmetic; eliminate all nonwinning dihedral orbits; and prove uniqueness inside the winning code by strong convexity and a quantitative KKT argument. The exact certificates cover $2^{15}$ normalized codes for $n=16$, $2^{31}$ normalized codes for $n=32$, and all $2^{64}$ half-codes for $n=64$, leaving respectively $16$, $96$, and $896$ survivors before orbit elimination. The accompanying source package contains the verifiers, recorded outputs, hashes, and separate computational cross-checks.

We investigate a computable and empirically implementable framework for continuous-time mean–variance optimal portfolio selection with random market coefficients. The market model is built on a tractable multifactor stochastic volatility structure, which captures state-dependent risk premia, stochastic volatility, and dynamic cross-asset dependence. The optimal control is characterized by stochastic Riccati equations. On the computational side, we design an iterative BSDE-based procedure to approximate the stochastic Riccati equation from above and below, where the initial upper and lower bounds are obtained by solving two linear BSDEs. We then apply a logarithmic transformation to remove the singularity in the SRE, and solve the resulting transformed equation using both Deep BSDE and DBDP methods. The linear BSDE bounds also provide effective initial-value estimates, improving the convergence speed and training stability of the Deep BSDE solver. Empirical experiments based on sector ETF data show that the proposed multifactor mean–variance strategy produces smooth target-return wealth dynamics, with improved drawdown control and downside-risk protection relative to benchmark strategies. These results demonstrate the practical potential of combining stochastic Riccati equations, neural BSDE solvers, and multifactor market modeling for dynamic asset allocation.

This study develops an analytical and decision framework for a pooled single-server diagnostic–treatment service receiving both first-time and referred patients. A first-time patient receives diagnosis followed immediately by one of $n$ treatment modes, whereas a referred patient enters directly into the prescribed treatment mode. For the two-treatment case, closed-form probability generating functions characterize the stationary queue-length distribution; for arbitrary $n$, a matrix-geometric representation provides a computational solution. The queueing analysis is then directed to capacity control decisions. The effective load is decomposed into mode-specific workload contributions, yielding a minimum service-rate increment for a prescribed utilization target to show that utilization-equivalent interventions need not be delay-equivalent. This motivates a distribution-aware control based on the stationary probability of severe congestion. Finally, the total cost function shown to be strictly convex, providing a global minimum for restricted one-mode interventions. Analytical results are cross-validated against the Pollaczek–Khintchine relations and discrete-event simulation.

This article studies the computation of the maximal controlled invariant set (MCIS) for neural network control systems (NNCSs) with a nominal plant model and a neural network residual model. An interval-based method is developed to construct a control-affine inclusion function for the NNCS dynamics and to exploit the induced hyperplane arrangement in the input space, which reduces the verification of controlled invariance from an infinite control search to a finite set of representative evaluations. Based on this finite characterization, verification-guided algorithms iteratively construct inner and outer approximations of the the MCIS, which are naturally parallelizable. Numerical examples demonstrate the effectiveness of the proposed approach.

We establish sharp asymptotic rates for the two Barzilai–Borwein (BB) rules on uniformly positive quadratics and local nonlinear problems. In finite dimensions, for either fixed rule and an arbitrary positive first step, the gradient root factor is bounded by $(b_0-a_0)/(b_0+a_0)$, where $[a_0,b_0]$ is the initially active spectral interval. Hence the worst trajectory factor is $c_H=(\kappa(H)-1)/(\kappa(H)+1)$. When $H$ has at least two distinct eigenvalues, matched initialization and a balanced endpoint trajectory attain this value. Under matched initialization, the same constant is the optimal uniform-envelope threshold. For bounded, self-adjoint, uniformly positive operators on Hilbert space, scalar spectral measures yield the corresponding active-support bound and optimal matched uniform-envelope threshold, including continuous endpoint spectrum. Finally, if the gradient is strictly Fréchet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $\gamma\in(c_*,1)$, where $c_*=(\kappa(A_*)-1)/(\kappa(A_*)+1)$, is a uniform local envelope rate for either pure BB rule. Every well-defined trajectory converging to the stationary point has error and gradient root factors at most $c_*$ and objective-gap root factor at most $c_*^2$. Over the class of objectives with prescribed distinct derivative endpoints $m_*<M_*$, matched endpoint trajectories for quadratic and $C^\infty$ genuinely nonquadratic examples in $\mathbb R^2$ attain these factors.

In sampling problems, gradient-based schemes such as Langevin Monte Carlo (LMC) mix faster than non-gradient-based methods, but their applicability is limited by access to the gradient of the target log-density. In practice, gradients are often unavailable and function evaluations are noisy, e.g., stochastic simulators or black-box simulators, so we propose LMC-SPSA with noise, which approximates the gradient of the target log-density using two noisy function evaluations per iteration. We prove, under noisy gradient estimates, that LMC-SPSA converges in distribution by proving the convergence in Wasserstein distance. Furthermore, we construct a diminishing step-size schedule that still drives the Wasserstein error bound to convergence, extending convergence guarantees beyond the constant-step setting. Further, we sharpen the dominant dimension dependence of the Wasserstein error from $O(p^4)$ to $O(p^2)$ (with $p$ denoting the dimension), and support this analysis with numerical results. We show that LMC-SPSA achieves $W_2$-accuracy $\varepsilon$ with total noisy-oracle complexity of $O(p/\varepsilon^2+\delta^2p^3/\varepsilon^3)$, where $\delta$ is the paired-noise level. This improves the noise-dependent accuracy scaling relative to the ZO-LMC method of Roy et al. We further establish asymptotically vanishing Wasserstein error as the number of iterations $\to\infty$ under diminishing step-size and perturbation sequences and derive an explicit convergence rate for a balanced schedule under noisy zeroth-order feedback. Empirical experiments are conducted to verify the performance of LMC-SPSA with noise. We provide an oracle-budget-matched comparison with the ZO-LMC method, showing smaller empirical sampling errors under the same function-evaluation budget.

Quadratic optimization becomes hard as soon as either the matrix in the quadratic form has an unfavorable curvature or the feasible set is discrete, combinatorial, or otherwise nonconvex. A complementary phenomenon is also well known in the signal-processing and optimization communities: when the matrix in the quadratic form has small rank, some hard-looking quadratic programs admit exact polynomial-time algorithms for fixed rank. We study the common positive-semidefinite geometry behind this phenomenon. If $Q=BB^\top$ is positive semidefinite, the objective depends on $x$ only through the rank-space shadow $y=B^\top x$. Every optimal shadow $y^*$ uniquely maximizes the linear functional defined by its own direction and satisfies a quantitative quadratic margin. Thus nonlinear optimality collapses to a low-dimensional, self-generated linear exposure direction. We call this the rank-collapse principle. The principle alone does not imply a finite candidate set: efficient exact optimization additionally depends on the projected or active geometry of the feasible family. We organize this distinction through projected-shadow scattering and active-structure collapse, relate it explicitly to established zonotope, convex-combinatorial, edge-skeleton, projected-normal-fan, and fixed-rank sparse-PCA methods, and derive tie-safe consequences for binary and finite-phase vectors, cardinality constraints, matroid bases, and sparse PCA. We also give directional-stability and approximately low-rank certificates, together with reproducible experiments. The paper's contribution is a unified, careful framework and a set of quantitative consequences, rather than a claim to originate the known fixed-rank tractability results that motivate it.

The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin's principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.

This paper develops a Hodge-theoretic feedback framework for flow-valued states, motivated by scalar parabolic thermostat problems with localized sensors and dual sources. On a connected graph, the edge space decomposes into a cut space and a harmonic cycle space. The ideal topological thermostat regulates the cut component toward a prescribed cycle-free transfer target and damps the harmonic component. For realizations through selected physical edge channels, the first Betti number gives the minimal sensing and actuation ranks required for exact compression of the full harmonic sector. Minimal rank does not imply local realization of the nonlocal Hodge projector; the remaining cut-harmonic blocks quantify spillover and forcing. For transmission networks, we formulate ideal and finite-channel nonlinear line-actuated swing closures and linearize the complete closed systems at an angle-cohesive synchronous equilibrium. With the actuator fixed, passive tangent swing motion preserves any pre-existing harmonic line-flow component. The ideal closure replaces this conservation law by exponential decay and appends a stable harmonic block without changing the reduced passive nodal-cut spectrum. Under the channel-rank conditions, finite realizations reproduce the harmonic compression, whereas autonomous harmonic decay and full reduced-state recovery require additional decoupling and Hurwitz conditions. A target-centred Bregman balance yields nonlinear dissipation for the ideal metric-compatible controller and for canonical colocated finite-channel feedback. Explicit theta-network, microgrid, ring, and synthetic IEEE 14-bus examples illustrate transfer regulation, cycle-flow damping, and localization-induced transients. Realistic device dynamics, constraints, and large-scale validation remain open.

The cosine measure of a set of vectors in $\mathbb{R}^n$ measures how well the set covers all directions in $\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\mathcal{C}^1$ function $f$ at a point $\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\mathbf{x}$. In this paper, we examine the question of what can be said when the function $f$ is non-differentiable or if it has a gradient equal to the zero vector. To examine these cases, we introduce the novel concept of the cosine measure of a function at a point. This value provides an infimum on the value of the cosine measure that a set of vectors requires to guarantee it contains a descent direction of the function at the point of interest. We present mathematical theory around this concept, including examples showing that the cosine measure of a smooth function can have any value in $[-1,1]$. We further present algorithms to compute the cosine measure of a function, and examples demonstrating the algorithm on smooth and nonsmooth functions. These results also shed light on the the cosine measure of infinite sets and nonconvex cones.

Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space. We study entropy-regularized discounted linear-quadratic (LQ) control. A Bellman verification argument shows that the unrestricted problem has a linear-Gaussian optimal policy, and the discounted-occupancy-weighted statewise Wasserstein gradient is tangent to this policy class. WPG therefore reduces exactly to a finite-dimensional ODE for the feedback gain and action covariance. We prove that this ODE is globally well posed and converges exponentially from every admissible initialization. For each fixed LQ problem, the exponent has a positive limit as the entropy temperature tends to zero and contains no perturbative factor of the form $\exp(-c/\tau)$, while retaining the usual dependence on the conditioning of the control problem.

We present AutoOPT, a domain-specific harness for end-to-end automation of optimization research. AutoOPT organizes the discovery of optimal first-order methods into four stages: numerical design through the BnB-PEP methodology; symbolic discovery of the analytic description and a convergence proof through frontier large language models (LLMs); formal verification in the Lean 4 proof assistant; and human interpretation and write-up. We demonstrate the framework on two case studies, each of independent interest. The first, lemniscate acceleration, is a new accelerated gradient method for minimizing the gradient norm of a smooth convex function: after $N$ gradient steps it reduces the squared gradient norm at the optimal $O(1/N^{4})$ rate, with a constant governed by the lemniscate constant $\varpi$, a classical elliptic-integral constant. The second is the analytic description of ITEM-f, a method previously known only numerically: for $L$-smooth, $\mu$-strongly convex minimization it contracts the function-value gap at an accelerated linear rate with a per-step factor $(1-\sqrt{\mu/L})^{2}$. The convergence theorems of both case studies are formalized and machine-checked in Lean 4.

Production planning in the manufacturing industry often relies on the use of optimization models, but defining an appropriate objective function can be a challenge. In practice, planners must balance competing goals, manage uncertainty, and account for qualitative business preferences that are difficult to quantify. As a result, many optimization models fail to match expert behavior, limiting trust and adoption. In this work, we propose a data-driven inverse optimization framework to infer the objective function implicitly captured in expert planners' decisions. We formulate the production planning problem as a mixed-integer linear program, where the unknown objective function is represented as a weighted sum of hypothesized cost terms. A suboptimality-loss-based inverse optimization method is then applied to learn the objective weights from historical production plans. The proposed approach is applied to a real industrial case provided by Dow, where the inferred weights reveal that avoiding inventory shortages and maintaining consistent cycle lengths dominate the planners' decision-making. Time- and product-dependent extensions further improve predictive accuracy and uncover evolving priorities. Expert interviews confirm the practical validity of these insights. Overall, this study shows that inverse optimization can transform tacit human expertise into interpretable models, enabling more accurate and trusted decision-support tools for complex industrial systems.

We consider generalized Nash equilibrium problems among $N$ players with convex quadratic costs and shared affine constraints, assuming only that the game's pseudogradient is merely monotone. We show that computing a variational generalized Nash equilibrium (v-GNE) is equivalent to solving a single convex quadratic program (QP) derived from the players' joint Karush–Kuhn–Tucker conditions. Building on this, we show that the regularization of such a QP yields an $\varepsilon$-approximated v-GNE with suboptimality vanishing linearly in the regularization parameter. Next, we propose an accelerated proximal-point scheme and an accelerated projected-gradient method, both attaining an $\mathcal O(1/k^2)$-approximated v-GNE at the $k$-th iteration. We also demonstrate that an invertible Jacobian of the game allows for reduction to a lower-dimensional QP. Theoretical analysis and numerical experiments show the proposed methods substantially outperform the existing approaches to solve monotone linear-quadratic v-GNE problems.

It is well known that the computational realization of Gibbs probability measures, $e^{-\mathbb{U}(\mathbf{x})}/Z$, plays a central role in sampling and optimization. In this paper, we introduce two types of dynamics that exhibit rapid convergence towards these Gibbs measures. The mechanism driving this rapid convergence is the enhanced dissipation associated with these transport-diffusion dynamics. Motivated by these enhanced dynamics, we design numerical algorithms for sampling from the target Gibbs measure. Finally, we provide the corresponding particle systems that may yield other effective numerical samplers.

In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.

We establish a finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems. Geometric supervision is used only during training: simulator state, proprioception, or state estimates with independently validated metric and directional error bounds supply observable-state distances and tangent directions, while deployment remains observation- and action-conditioned. We introduce an encoder-only local–global metric hinge that enforces directional resolution and separated-state discrimination. Under regular observable-factor, coverage, finite-capacity approximation, and uniform $C^{1,1}$ hypotheses, a computable one-sided regularization regime has a strong selection property: with high probability, every approximate empirical minimizer is simultaneously pointwise co-Lipschitz and uniformly approximately semiconjugate to the controlled dynamics. Approximation, sampling, and optimization errors remain explicit and separate. Norm-constrained tensor-product B-spline classes constructively realize the approximation hypotheses, and the interpolation exponent converting mean residual control into a uniform bound is sharp. A modular deterministic corollary transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer guarantees, while a validated finite-net result enables sharper model-specific certification. Controlled experiments isolate collapse and folding, quantify the analytic certificate's reserve, and demonstrate the control benefit of restored metric resolution. The principal contribution is a complete finite-sample implication from approximate empirical optimization to metric faithfulness, uniform controlled dynamics, and reliable planning for the same learned model.

This paper studies the inverse reinforcement learning (RL) problem for linear-quadratic mean-field (MF) social optimization. The considered system features multiplicative noise and indefinite cost weights, which violate standard convexity assumptions and pose analytical challenges. The goal is to recover unknown social cost weights from expert demonstrations and reproduce the optimal control policies. This requires solving coupled stochastic algebraic Riccati equations and Lyapunov equations with unknown system dynamics. To this end, we first propose a model-based inverse RL algorithm with two sequential loops that separately handle individual and MF dynamics, and we prove its convergence and closed-loop stabilizability. Moreover, we characterize the non-uniqueness of the recovered cost weights. To eliminate reliance on system dynamics, we develop a model-free inverse RL algorithm using integral RL and least-squares identification, which requires only measured trajectory data satisfying mild rank conditions. Finally, numerical simulations validate the effectiveness of the proposed approaches.

Sequence convergence to a boundary Karush–Kuhn–Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi–Dirac entropy, and power kernels on polyhedron.

When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.

This paper presents a novel trajectory optimization method for nonlinear l1-optimal control problems in which the control horizon is treated as a free variable, allowing both the control duration and the l1-norm of the control input to be evaluated. Unlike traditional approaches that minimize the l1-norm of the input under a fixed control horizon, the proposed method treats both the control input and control horizon as design variables, enabling joint optimization of energy and temporal efficiency. The method extends a Newton-based algorithm to accommodate this objective, leveraging gradient information with respect to both variables. This formulation enables one to find high-fidelity optimal trajectories without pre-specifying the control horizon. The effectiveness and robustness of the method are demonstrated through numerical simulations, in which the proposed algorithm recovers the analytical solution to the spacecraft Hohmann transfer, and determines an optimal control horizon for an Earth-Moon transfer.

We develop a certified, scalable approximation for high-dimensional Wasserstein distributionally robust portfolio optimization. For expected-utility maximization under order-one Wasserstein ambiguity, standard duality yields a semi-infinite convex program. For long-only portfolios with box support under the one-norm ground metric, an exact sample-specific vertex reformulation provides an exponential-size computational benchmark. We then majorize the utility by supporting hyperplanes and dualize the support subproblems, obtaining a finite hyperplane–dual formulation over compact polyhedral supports. Under the one-norm ground metric and polyhedral portfolio constraints, this formulation is a polynomial-size linear program. The uniform utility-approximation error bounds both the robust-value error and the near-optimality gap for the original robust problem. Experiments validate the certified approximation and demonstrate monthly 476-asset rebalancing and computational scalability to 1,000 assets.