Papers

Branch model predictive control optimizes multiple future trajectories coupled through shared decisions, with computational demands increasing as the number of scenarios and prediction horizon grow. We present a GPU-accelerated direct linear solver for branch MPC formulations in which all trajectories share a single root decision node and evolve independently thereafter. By operating at the linear-algebra level, the solver provides a reusable backend for multiple optimization algorithms whose reduced systems have the required symmetric positive-definite structure. The solver exploits two levels of parallelism: across scenarios and along each prediction horizon. A tailored variable ordering enables horizon-parallel Cholesky factorization while preserving a single root-tail coupling block per scenario in the factor. Numerical experiments demonstrate substantial speedups over state-of-the-art sparse direct solvers, achieving factorization speedups of up to 6.0$\times$ over cuDSS and 27.6$\times$ over eight-thread PARDISO, with triangular solve speedups of up to 3.5$\times$ and 15.8$\times$, respectively.

Many iterative algorithms rely on bootstrapping. A variable is updated using a second, frozen copy as a target, which is periodically replaced with the updated variable. Majorize-minimize and inexact proximal-point methods share this structure, as does temporal-difference (TD) learning. However, existing convergence guarantees for scenarios that combine sampled updates with targets refreshed only every $K$ steps rely on the specific structure of the update, such as linear approximation or gradient-based inner steps, and on uniformly bounded sampling error. We instead model the sampled update as a stochastic operator on the parameter space, which reduces the analysis to a contraction argument that needs no gradient structure and allows the sampling error to grow with the iterates. Within this framework, we derive a finite-time bound for i.i.d. samples and any target-update period $K$. We show that the iterates converge geometrically in root mean square to a ball around the fixed point, provided the sensitivity to the frozen target is smaller than the contraction slack of the inner map. Existing deterministic frozen-target contraction and stochastic-gradient-type bounds follow as special cases of our framework, and simulations of TD learning reproduce the predicted contraction rate and scaling of the error floor with the step size.

Many real-world optimization problems rely on expensive simulations or experiments, making the efficient use of available data essential. Multi-fidelity optimization of high-dimensional black-box functions subject to black-box constraints is increasingly relevant as the cost of objective evaluations continues to rise in applications such as machine learning, engineering, and control. To our knowledge, no existing method simultaneously addresses high-dimensionality, black-box constraints, an arbitrary number of fidelity levels, and non-nested sampling. In this work, we extend the Scalable Constrained Bayesian Optimization method to the multi-fidelity setting, resulting in the MF-SCBO method. The proposed approach is evaluated on standard benchmark functions as well as challenging problems. The experimental results demonstrate that MF-SCBO generally achieves better convergence than both the single-fidelity SCBO and the other multi-fidelity method considered in this high-dimensional and constrained settings.

We develop randomized branch methods for finding Bouligand-stationary (B-stationary) points of linear and quadratic programs with complementarity constraints (LPCCs and QPCCs) using only linear programming subproblems. The methods exploit the finite-union geometry of the feasible set and search for first-order descent on randomly selected compatible branches. For LPCCs, an exact method optimizes over sampled branches, while an inexact simplex method can accept an improving branch-feasible vertex before solving the sampled penalty problem to optimality. For QPCCs, including problems with indefinite quadratic objectives, branch quadratic programs are replaced by linearized trust-region subproblems; a ratio test ensures actual decrease, and thresholded sampling detects branches that emerge only at accumulation points. Under the stated assumptions, the LPCC methods stabilize after finitely many changes at B-stationary points almost surely. For the QPCC method, finite termination yields a B-stationary point, and every accumulation point of an infinite run is B-stationary almost surely. Experiments on bilevel-induced instances, instances arising from inverse quadratic programming, and sparse affine generalized Nash equilibrium instances, together with 129 MacMPEC embedding tests, show that the methods return points with competitive objective quality and runtimes on large-scale complementarity systems.

This paper investigates cost-sensitive online window size selection for portfolio management under changing market conditions. Specifically, we propose a two-level framework that constructs portfolios using candidate window sizes and dynamically aggregates them through online learning. By treating candidate window sizes as “experts,” we dynamically update their aggregation weights using turnover-inclusive losses. Moreover, we derive finite-horizon cost-sensitive tracking-regret bounds that account for turnover of the aggregated portfolio, with static regret as a special case. Under bounded losses and cost rates, suitably tuned Fixed Share achieves asymptotically no tracking regret for sublinear switching budgets, with Hedge covering the static case.

Adaptive algorithms increasingly make decisions while reshaping the dynamics that generate their future data. We establish a shrinking-tube concentration bound for projected stochastic approximation driven by an adaptive Markov chain. The bound guarantees, with high probability, that every iterate after a chosen time remains within a tolerance around the target that tightens over time. The probability of any exit after the chosen time admits a polynomially decaying upper bound, and a matching lower bound shows that its polynomial exponent cannot be improved in general under finite second moments. The result therefore identifies a sharp tradeoff between how quickly the tolerance shrinks and how rapidly the probability of any future exit decreases. We also extend the analysis to recursions with additional martingale-difference noise and predictable bias, showing how growth in the martingale-difference noise scale slows the decay of the exit-probability bound while predictable bias restricts the admissible tube shrinkage. The proof combines backward kernel replacement, a finite-time mean-squared-error bound, and a blockwise maximal first-exit argument. We apply the theory to inventory learning with stockout-dependent demand and fixed stockout costs, and quantify how numerical gradient accuracy affects the all-future reliability of the resulting policies.

We propose a discrete-time feedback linearization control approach to solve nonsmooth linearly constrained composite optimization problems. By using proximal operators, we construct a dynamical system whose equilibria correspond to the stationary points of the optimization problem. Interpreting the Lagrange multipliers as control inputs, we employ feedback linearization control to steer the obtained dynamical system to a stable equilibrium. We analyze the convergence of the resulting closed-loop system both in the strongly convex setting and under the Polyak-Lojasiewicz condition. Finally, we illustrate the applicability of the approach through numerical experiments.

Enhancing the robustness of deployed networks against failures and disruptions is critical for reliable operation. This requires deciding which new links to install and how strongly to weight them under limited resources. We study this problem through the Kirchhoff index, or total effective resistance, a spectral measure of global connectivity. The resulting augmentation problem couples discrete candidate-edge selection with continuous weight allocation under heterogeneous per-unit deployment costs, a total budget, and an exact-cardinality constraint. For a fixed weighted base graph, this yields a mixed-integer formulation and a semidefinite relaxation whose optimum lower-bounds the mixed-integer optimum. We cast the relaxation as a cone program and solve it numerically using a homogeneous self-dual embedding and first-order operator splitting. Feasible discrete designs are recovered through rounding-and-repair procedures and assessed by a posteriori gap estimates relative to the numerical semidefinite program (SDP) benchmark. As a scalable alternative, we develop an exact-$k$, budget-feasible greedy heuristic built on rank-one Laplacian updates and biharmonic-distance caching, and interpret its progress through a Bellman value-to-go benchmark with a conservative spectral lower bound on the local policy ratio. Experiments on synthetic and real infrastructure networks across graph sizes, budgets, weight distributions, and cost regimes show that, under fixed budgets, distance-proportional costs limit the achievable resistance reduction and shift installed conductance toward shorter links relative to uniform per-unit costs.

We present a tightened convergence analysis of clipped gradient descent on $(L_0, L_1)$-smooth functions, with quantitative constants. Building on the ideas of Koloskova et al (2023), we refactor several case disjunctions to reveal the central role of a control of the bias derived from fundamental properties of $\ell_2$-projection, simplifying proofs. We also extend the domain of validity from $\eta \leq 1 / (9 \beta)$ to $\eta < 1 /\beta$ where $\beta = L_0 + c L_1$ for clipping constant $c$, which matches the more traditional analysis of smooth functions. We strengthen the convergence criterion from $\left( \min_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ to $\left( \frac{1}{T} \sum_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ with matching speed, and lower the final achievable loss from $\mathcal{O}(\min(\sigma^2/c, \sigma))$ to the more precise $6 \min(\sigma^2 /c, 3 \sigma)$.

Equilibrium modeling for power markets usually assumes convexity of each players' optimization problem. Although the importance of accounting for fixed generation costs and start-up costs, minimum generation levels and/or minimum up-time and down-time restrictions in production scheduling is widely acknowledged, such modeling does not allow for discrete decisions. This paper considers unit commitment constrained Nash equilibria. First, we derive novel optimality conditions tailored for the mixed-integer convex programming problem of joint unit commitment and economic dispatch of a self-scheduling profit-maximizing producer. Next, we use these to formulate a Nash equilibrium as a mixed-integer complementarity problem, which facilitates the development of a procedure to obtain multiple equilibria. The approach can be adapted to both perfectly and imperfectly competitive market settings. While an equilibrium may not always exist, we give sufficient conditions under which a Cournot-Nash equilibrium can be obtained by mixed-integer convex programming under the standard assumption of an affine inverse demand curve and convex generating costs, and use this to establish existence. A case study demonstrates the feasibility of using optimisation to obtain a Cournot-Nash equilibrium for unit commitment constrained market clearing. Our results confirm that ignoring unit commitment creates significant welfare losses, although these vary substantially across equilibria.

Convergence of the bilevel policy iteration algorithm for the value function, satisfying the Hamilton-Jacobi-Isaacs (HJI) equation, is analyzed in the presence of control constraints. Both first and second-order HJI equations corresponding to the deterministic and stochastic systems are considered. For numerical tests, a semi-implicit upwind scheme for backward HJI PDEs is applied and a thorough comparison between the solutions to first and second-order Hamilton-Jacobi-Bellman (HJB) equations is presented for both the unconstrained and the constrained control cases.

We consider control-affine optimal control problems on the torus, where the dynamics and cost functions are only accessed through samples. Starting from a weak formulation of such problems, we derive a dual, a primal, and a primal-dual formulation, compatible with stochastic optimization. We show convergence of stochastic first-order methods to the optimal value under generic conditions. In addition, we introduce a computable metric that upper-bounds the performance of suboptimal controllers produced during optimization, under additional regularity assumptions. Preliminary results show that the method can efficiently solve a simple control problem. Finally, we discuss conditions for the boundedness of the optimal occupation measure, a key assumption for the primal and primal-dual approaches.

This paper studies the time optimal control problem for the heat equation with singular inverse-square potentials under the Hardy critical condition. We first establish an observability inequality for the singular parabolic equation from general space-time measurable sets of positive Lebesgue measure. Rather than depending on the still-unknown Lebeau-Robbiano spectral inequality for the underlying singular operator, our proof combines Carleman-based observability results over open cylinders, real-analyticity estimates of solutions away from the singular origin, propagation of smallness estimates for real-analytic functions, and a telescoping-series technique. Using this observability result, we derive the null-controllability with controls supported on measurable subsets. Finally, we prove that the corresponding time optimal control is unique and obeys the bang-bang property almost everywhere over the control domain.

This paper investigates the null controllability of the complex cubic Ginzburg–Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.

We consider a stochastic parabolic equation on the $n$-dimensional flat torus $\mathbb{T}^n=(\mathbb{R}/(2\pi\mathbb Z))^n$, $n\in\mathbb N$, whose only control acts in the drift. The multiplicative-noise coefficient is adapted and may depend jointly on the sample point, time, and space. Its spatial profiles are assumed to be real analytic with a uniform positive radius of analyticity, while the corresponding analytic norm is only required to be square integrable in time, uniformly along sample paths. We prove a state-only observability inequality with an $L^1$-in-time mean-square observation norm and deduce null controllability from every measurable spatial set of positive measure by one adapted drift control. The same one-time interpolation estimate also yields approximate controllability to arbitrary square-integrable random terminal targets. The proof is based on a direct analytic energy estimate for the adjoint state, followed by propagation of smallness and a telescoping argument.

We establish an averaging principle for a class of multiscale mean-field forward-backward stochastic differential equations and identify several novel phenomena that are absent from classical fast-slow systems. In contrast with classical fast-slow systems, the effective dynamics cannot in general be obtained by simply freezing deterministic slow parameters and averaging against the invariant measure of the resulting fast equation. The appropriate averaging object is instead provided by a frozen fast dynamics in a random environment and its associated conditional invariant measures, which retain the coupling between the slow state and its distribution. The forward-backward structure creates a further obstruction: local averaging estimates need not remain stable when propagated over an arbitrary time horizon. We identify a uniform restart stability condition for the averaged system under which this obstruction can be overcome. Using a joint lifted semigroup for the state-law dynamics, together with a two-scale discretization and a Gordin-type decomposition, we prove strong averaging for both the forward and backward components with optimal convergence rate $O(\varepsilon^{1/2})$. As an application, we apply the general theory to a class of mean-field stochastic control problems and develop an efficient algorithm for solving such mean-field control problems.

We establish a randomized first-order lower bound for finding an $\epsilon$-stationary point, $\|\nabla f(x)\|\le \epsilon$, of a nonconvex function with initial gap at most $\Delta$, $L_1$-Lipschitz gradient, and $L_2$-Lipschitz Hessian. Each oracle call returns the exact function value and gradient. Let $Q_{\mathrm{rand},\mathrm{FO}}^{\infty}(\epsilon;\Delta,L_1,L_2)$ denote the minimax number of calls, maximized over finite dimensions, for arbitrary adaptive randomized algorithms with per-instance success probability at least $2/3$. In the regime $\epsilon\lesssim L_1^2/L_2$ and $\Delta L_2^{1/2}\epsilon^{-3/2}\gtrsim 1$, we prove $ Q_{\mathrm{rand},\mathrm{FO}}^{\infty}(\epsilon;\Delta,L_1,L_2) \ge c\,\Delta L_1^{1/2}L_2^{1/4}\epsilon^{-7/4}, $ where $c>0$ is an absolute constant. This matches the deterministic restarted accelerated-gradient upper bound of Li and Lin (2023) and extends the sharp deterministic lower bound of Zhou (2026) to unrestricted randomized algorithms. Thus, randomization does not improve the high-dimensional worst-case query rate. The construction keeps quadratic curvature visible while revealing the forcing directions sequentially. Large eigenspaces limit preprocessing, and a scheduled-disclosure coupling makes each fresh direction incur a separate cost.

Two phase-optimized multisine peak factor minimization algorithms are presented and evaluated. The first algorithm minimizes peak factor by iteratively clipping the peaks of generated multisine signals. The second algorithm optimizes peak factor indirectly through minimization of an approximation of the infinity norm of the multisine. Algorithm performance was evaluated as a function of different signal properties, including the number of harmonics, harmonic spacing, and number of snow harmonics (extra harmonics included for further reduction of the peak factor). The two algorithms are compared against results obtained by minimizing peak factor directly using a simplex algorithm, which has been a common approach when designing phase-optimized multisines for system identification flight tests. Sample results show that the clipping and infinity norm algorithms produced multisine signals with comparable peak factors that were lower than that of the simplex algorithm. However, the clipping algorithm runs an order of magnitude faster than the other two algorithms, which also makes it practical to repeat the algorithm multiple times to achieve even lower peak factors.

The approximation of nonlinear functions by piecewise linear functions is a tool commonly used when dealing with mixed-integer nonlinear problems. Typically, by replacing nonlinearities by piecewise linear functions one can transform the problem into a mixed-integer linear problem, which may be substantially easier to solve. However, using approximate functions can produce solutions that are infeasible for the original problem or far from optimal. To control these errors it is useful to bound the error created during the function approximation process. Moreover, obtaining a piecewise linear function with few pieces usually results in an easier to solve mixed-integer linear problem. This leads us to study the Corridor Fitting Problem. It consists in building a piecewise linear function with the minimum number of pieces which approximates a nonlinear function given a bound on the approximation error on each point of the domain. The Corridor Fitting Problem has primarily been addressed for univariate functions or via heuristic approaches for multivariate functions. Notably, for the latter setting, no exact algorithms or established relaxations are currently known. In this work, we explore this aspect and propose exploitable relaxations of the Corridor Fitting Problem in Rm based on a discretization of the domain. We show that a structure of hypergraph coloring problem is induced by the discretization of the domain. We define four relaxations making use of this hypergraph coloring problem. We provide new best upper bounds for the classical instance set in R2 and we derive the first lower bounds for these instances, closing more than a third of the instances from the literature.

This paper proposes a fuel-optimal boost-back guidance algorithm for reusable launch vehicles using successive convexification (SCvx). The guidance problem is formulated as a free-final-time optimal control problem with a terminal instantaneous impact point (IIP) constraint. This constraint depends only on the burnout position and velocity and enforces that the predicted ballistic impact point coincides with the target under a spherical-Earth, central-gravity model. By handling the coast analytically, the formulation confines trajectory discretization to the powered phase, reduces the problem size, and explicitly represents the bang-off structure. It also avoids the accumulation of discretization defects over the long coast and the need to resolve the burn-coast transition on a single-phase grid. Two terminal constraint formulations are considered: the closed-form Keplerian IIP and the eccentric-anomaly-based F&G solution. Their Jacobians are evaluated using complex-step differentiation. The algorithm is validated in a return-to-launch-site case study based on the Falcon 9 CRS-10 mission. Comparisons with a single-phase full-trajectory formulation, a closed-form guidance law incorporating the flight path angle rate, and an offline trajectory optimization benchmark assess the trade-off between computational cost and optimality. The results demonstrate near-optimal propellant consumption with reduced computational cost.

Longitudinal spacing models describe the speed–spacing states that vehicles maintain, while longitudinal controllers regulate vehicle motion through spacing and speed errors. Less explicit is how one spacing state should be connected to another through a finite, physically meaningful vehicle transition. This paper develops such a connection through r-Safety and Adaptive Kinematic Smoothing (AKS). The r-Safety relation represents speed-dependent spacing through a braking-normalized reserve, whose change, together with leader motion, determines the follower displacement required during a transition. AKS then converts this displacement into a low-dimensional analytical trajectory with consistent position, speed, spacing, and acceleration. Experiments using controlled platoons and NGSIM trajectories identify r-Safety values of 0.307 and 0.249–0.281, respectively, and show that AKS can both represent observed transitions and generate prospective references under prescribed endpoint states, transition horizons, and leader motion. In same-controller ablations, adding AKS reduces integrated acceleration effort by 21–47% and integrated squared jerk by 73–81% across the controlled and naturalistic datasets. FHWA ACC/CACC trajectories further provide an empirical illustration of finite-horizon execution under automated following, showing how spacing adjustment can continue beyond the prescribed speed transition. The framework therefore provides a direct path from a speed–spacing relation to an analytical finite transition and its use as a longitudinal control reference.

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.

These are extended notes based on a series of lectures given by the author at the Institut de Mathematiques de Toulouse in June 2025. The goal is to present classical problems as well as the most recent developments in spectral geometry of the Laplace operator in Euclidean domains, subject to Dirichlet, Neumann and Robin boundary conditions. Among the topics covered, there are embedded eigenvalues in unbounded domains, spectral isoperimetric inequalities, nodal-line and hot-spots conjectures, inverse problems of hearing the shape of a drum and geometrically induced eigenvalues and Hardy-type inequalities in curved tubes.

We study dynamic pricing problems under inter-temporal price constraints. We have resources with limited capacities. At each time period, we decide which products to make available and what prices to charge for the available products. The sale probability for a product depends on its price. If we make a sale for a product, then we collect a revenue reflecting the price and consume the capacities of a combination of resources. We work with two types of inter-temporal constraints. In price monotonicity, the prices charged for a product at different time periods have to be monotone. In promotion fatigue, we can discount a product at most once over each time interval of a fixed length. Computing the optimal policy is intractable. We use fluid approximations to construct policies. Traditionally, policies from fluid approximations make randomized decisions at each time period by following an optimal solution to the fluid approximation, but such randomized decisions easily violate price monotonicity or promotion fatigue constraints. We develop policies that sample price paths according to an optimal solution to the fluid approximation, while satisfying the inter-temporal constraints. Letting $c_{\min}$ be the smallest initial capacity of a resource and $L$ be the maximum number of resources used by a product, our policies have a performance guarantee of $\max\Big\{ \frac{1}{8L}, \, \frac{1}{2} - \sqrt{\frac{\log c_{\min}}{2 \,c_{\min}}} - \frac{L}{c_{\min}}\Big\}$. Thus, under large resource capacities, our policies are guaranteed to obtain at least half of the optimal total expected revenue. The latter performance guarantee is tight in the sense that no policy can, in general, obtain more than half of the optimal objective value of the fluid approximation even under large resource capacities. We unify our approach to open the path for extensions to other inter-temporal price constraints.

For a convex body $K\subset\mathbb{R}^n$, let $w(K)$ denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace $H$ whose boundary passes through the center of the maximizing ellipsoid satisfies \[ w(K\cap H)\le\frac{\sqrt e}{2}\,w(K). \] The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.

Terminal intercept-angle control against a maneuvering target can substantially increase the required missile acceleration, potentially leading to saturation and interception failure unless acceleration limits are explicitly addressed. The engagement is therefore formulated as a linear-quadratic optimal-control problem with bounded acceleration commands. Polynomial approximations of the line-of-sight projection coefficients are used to better represent the nonlinear engagement geometry and estimate the time-to-go. The bounded optimal command is derived over saturated and unsaturated arcs, whose switching times are computed at each guidance step. The guidance law is derived for arbitrary linear missile dynamics and implemented for zero-order missile dynamics. For the zero-order model, the conditions under which the terminal demands can be met are derived in closed form, yielding the minimum and maximum reachable commanded terminal intercept angles. Performance is evaluated in nonlinear simulations. Compared with its unconstrained counterparts, the bounded formulation yields substantially smaller miss distances and terminal-angle errors when saturation is encountered. Unlike corresponding bounded miss-only guidance laws, the proposed law does not reduce to its unconstrained counterpart for minimum-phase missile dynamics because the acceleration command can saturate near the end of challenging engagements. The bounded law anticipates this saturation and compensates through earlier maneuvers.

Control barrier functions (CBF) are a popular safety filter to ensure safety for nonlinear dynamical systems. However, when the system is subject to uncertainties and disturbances, this requires the use of robust variants of CBFs, which can be difficult to construct and can be overly conservative, especially for high-dimensional systems under input constraints. In this work, we propose a new approach to solve these challenges by introducing Least-Effort Adversarial Potentials (LEAP), a certificate that quantifies the robustness of a given state against disturbances in terms of the effort required by the disturbance to cause failure. We show that LEAP is a CBF for the undisturbed system, but can also be used to construct a safety filter that is robust to disturbances whose cumulative effort is bounded. We propose a method for constructing LEAPs with on-policy deep reinforcement learning. Next, we demonstrate LEAPs in simulation on a variety of multi-agent systems with disturbances and uncertainties. Finally, hardware experiments on a quadruped and quadrotors validate that LEAPs are well suited to tackle the disturbances and uncertainties from real-world robotic systems.

We introduce the effective curvature dimension of a smooth DC decomposition, defined as the dimension of the span of the Hessian ranges of its first convex component. We prove that the smallest attainable effective dimension over all smooth DC decompositions equals the minimum rank of a positive semidefinite matrix that uniformly majorizes the Hessian of the DC objective. Hence, an optimal dimension is always attained by a quadratic convexifier. We further characterize feasible curvature-free subspaces through generalized Schur-complement criteria, revealing an obstruction caused by cross-curvature.As an algorithmic consequence of the effective dimension framework, we show that the number of globally solved lower model subproblems depends on the effective dimension rather than the ambient dimension. Finally, for quadratic first components, we identify the minimum number of tangent supports required for uniform approximation with an internal covering number and obtain a matching dimensional exponent under a nondegeneracy condition.

We establish whole-sequence convergence of the primal iterates of the smoothing-based full-splitting proximal subgradient method (S-FSPS) of Boţ, Li, and Tao (SIAM J. Optim., 35 (2025), pp.~2623–2653) with a prescribed, nonsummable sequence of vanishing smoothing parameters. The challenge is that each iteration uses a different smoothed model. We address this by constructing fixed lifted potentials, deriving compatible sufficient-decrease and relative-error estimates, and applying a scaled Kurdyka–Łojasiewicz (KL) finite-length argument that controls the residuals caused by changes in the smoothing parameter. Under a KL assumption on the corresponding fixed lifted potential and a weighted summability condition on these residuals, the primal trajectory has finite length. For power schedules $\gamma_k=(k+k_0)^{-\beta}$, with $k_0\geq1$ and $1/2<\beta\leq1$, the summability condition holds when the lift exponent is sufficiently large. Consequently, the primal sequence converges to an exact limiting lifted stationary point without imposing full-row-rank assumptions on the linear operators. As a corollary, we establish whole-sequence convergence for a variable-smoothing full-splitting projected-gradient method in nonconvex nonsmooth composite optimization. Examples distinguish primal convergence from convergence of the auxiliary dual variables and illustrate the role of nonsummability in guaranteeing exact stationarity.

We study supinf problems for perturbations of a given function of two variables by continuous bounded functions in the underlying spaces and the properties of the associated solution mapping. We provide conditions on the initial data such that most (in the Baire category sense) of the functions from the perturbation space secure existence and uniqueness of the solution of the supinf problem for the corresponding perturbed functions. In this way we obtain generic variational principles for such problems.