CyberSec Research

We introduce a semidefinite relaxation for optimal control of linear systems with time scaling. These problems are inherently nonconvex, since the system dynamics involves bilinear products between the discretization time step and the system state and controls. The proposed relaxation is closely related to the standard second-order semidefinite relaxation for quadratic constraints, but we carefully select a subset of the possible bilinear terms and apply a change of variables to achieve empirically tight relaxations while keeping the computational load light. We further extend our method to handle piecewise-affine (PWA) systems by formulating the PWA optimal-control problem as a shortest-path problem in a graph of convex sets (GCS). In this GCS, different paths represent different mode sequences for the PWA system, and the convex sets model the relaxed dynamics within each mode. By combining a tight convex relaxation of the GCS problem with our semidefinite relaxation with time scaling, we can solve PWA optimal-control problems through a single semidefinite program.

The "avoid - shift - improve" framework and the European Clean Vehicles Directive set the path for improving the efficiency and ultimately decarbonizing the transport sector. While electric buses have already been adopted in several cities, regional bus lines may pose additional challenges due to the potentially longer distances they have to travel. In this work, we model and solve the electric bus scheduling problem, lexicographically minimizing the size of the bus fleet, the number of charging stops, and the total energy consumed, to provide decision support for bus operators planning to replace their diesel-powered fleet with zero emission vehicles. We propose a graph representation which allows partial charging without explicitly relying on time variables and derive 3-index and 2-index mixed-integer linear programming formulations for the multi-depot electric vehicle scheduling problem. While the 3-index model can be solved by an off-the-shelf solver directly, the 2-index model relies on an exponential number of constraints to ensure the correct depot pairing. These are separated in a cutting plane fashion. We propose a set of instances with up to 80 service trips to compare the two approaches, showing that, with a small number of depots, the compact 3-index model performs very well. However, as the number of depots increases the developed branch-and-cut algorithm proves to be of value. These findings not only offer algorithmic insights but the developed approaches also provide actionable guidance for transit agencies and operators, allowing to quantify trade-offs between fleet size, energy efficiency, and infrastructure needs under realistic operational conditions.

We develop two classes of variance-reduced fast operator splitting methods to approximate solutions of both finite-sum and stochastic generalized equations. Our approach integrates recent advances in accelerated fixed-point methods, co-hypomonotonicity, and variance reduction. First, we introduce a class of variance-reduced estimators and establish their variance-reduction bounds. This class covers both unbiased and biased instances and comprises common estimators as special cases, including SVRG, SAGA, SARAH, and Hybrid-SGD. Next, we design a novel accelerated variance-reduced forward-backward splitting (FBS) algorithm using these estimators to solve finite-sum and stochastic generalized equations. Our method achieves both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ convergence rates on the expected squared norm $\mathbb{E}[ \| G_{\lambda}x^k\|^2]$ of the FBS residual $G_{\lambda}$, where $k$ is the iteration counter. Additionally, we establish, for the first time, almost sure convergence rates and almost sure convergence of iterates to a solution in stochastic accelerated methods. Unlike existing stochastic fixed-point algorithms, our methods accommodate co-hypomonotone operators, which potentially include nonmonotone problems arising from recent applications. We further specify our method to derive an appropriate variant for each stochastic estimator -- SVRG, SAGA, SARAH, and Hybrid-SGD -- demonstrating that they achieve the best-known complexity for each without relying on enhancement techniques. Alternatively, we propose an accelerated variance-reduced backward-forward splitting (BFS) method, which attains similar convergence rates and oracle complexity as our FBS method. Finally, we validate our results through several numerical experiments and compare their performance.

In this paper, we present Quantum-Inspired Model Predictive Control (QIMPC), an approach that uses Variational Quantum Circuits (VQCs) to learn control polices in MPC problems. The viability of the approach is tested in five experiments: A target-tracking control strategy, energy-efficient building climate control, autonomous vehicular dynamics, the simple pendulum, and the compound pendulum. Three safety guarantees were established for the approach, and the experiments gave the motivation for two important theoretical results that, in essence, identify systems for which the approach works best.

We consider the Mathematical Program with Complementarity Constraints (MPCC). One of the main challenges in solving this problem is the systematic failure of standard Constraint Qualifications (CQs). Carefully accounting for the combinatorial nature of the complementarity constraints, tractable versions of the Mangasarian Fromovitz Constraint Qualification (MFCQ) have been designed and widely studied in the literature. This paper looks closely at two such MPCC-MFCQs and their influence on MPCC algorithms. As a key contribution, we prove the convergence of the sequential penalisation and Scholtes relaxation algorithms under a relaxed MPCC-MFCQ that is much weaker than the CQs currently used in the literature. We then form the problem of tuning hyperparameters of a nonlinear Support Vector Machine (SVM), a fundamental machine learning problem for classification, as a MPCC. For this application, we establish that the aforementioned relaxed MPCC-MFCQ holds under a very mild assumption. Moreover, we program robust implementations and comprehensive numerical experimentation on real-world data sets, where we show that the sequential penalisation method applied to the MPCC formulation for tuning SVM hyperparameters can outperform both the Scholtes relaxation technique and the state-of-the-art derivative-free methods from the machine learning literature.

This document presents an Integer Linear Programming (ILP) approach to optimize pedestrian evacuation in flood-prone historic urban areas. The model aims to minimize total evacuation cost by integrating pedestrian speed, route length, and effort, while also selecting the optimal number and position of shelters. A modified minimum cost flow formulation is used to capture complex hydrodynamic and behavioral conditions within a directed street network. The evacuation problem is modeled through an extended graph representing the urban street network, where nodes and links simulate paths and shelters, including incomplete evacuations (deadly nodes), enabling accurate representation of real-world constraints and network dynamics.

The reliability of a system can be improved by the addition of redundant elements, giving rise to the well-known redundancy allocation problem (RAP), which can be seen as a design problem. We propose a novel extension to the RAP called the Bi-Objective Integrated Design and Dynamic Maintenance Problem (BO-IDDMP) which allows for future dynamic maintenance decisions to be incorporated. This leads to a problem with first-stage redundancy design decisions and a second-stage sequential decision problem. To the best of our knowledge, this is the first use of a continuous-time Markov Decision Process Design framework to formulate a problem with non-trivial dynamics, as well as its first use alongside bi-objective optimization. A general heuristic optimization methodology for two-stage bi-objective programmes is developed, which is then applied to the BO-IDDMP. The efficiency and accuracy of our methodology are demonstrated against an exact optimization formulation. The heuristic is shown to be orders of magnitude faster, and in only 2 out of 42 cases fails to find one of the Pareto-optimal solutions found by the exact method. The inclusion of dynamic maintenance policies is shown to yield stronger and better-populated Pareto fronts, allowing more flexibility for the decision-maker. The impacts of varying parameters unique to our problem are also investigated.

First, we analyze the discrete Monge--Kantorovich problem, linking it with the minimization problem of linear functionals over adjoint orbits. Second, we consider its generalization to the setting of area preserving diffeomorphisms of the annulus. In both cases, we show how the problem can be linked to permutohedra, majorization, and to gradient flows with respect to a suitable metric.

We prove a novel and general result on the asymptotic behavior of stochastic processes which conform to a certain relaxed supermartingale condition. Our result provides quantitative information in the form of an explicit and effective construction of a rate of convergence for this process, both in mean and almost surely, that is moreover highly uniform in the sense that it only depends on very few data of the surrounding objects involved in the iteration. We then apply this result to derive new quantitative versions of well-known concepts and theorems from stochastic approximation, in particular providing effective rates for a variant of the Robbins-Siegmund theorem, Dvoretzky's convergence theorem, as well as the convergence of stochastic quasi-Fej\'er monotone sequences, the latter of which formulated in a novel and highly general metric context. We utilize the classic and widely studied Robbins-Monro procedure as a template to evaluate our quantitative results and their applicability in greater detail. We conclude by illustrating the breadth of potential further applications with a brief discussion on a variety of other well-known iterative procedures from stochastic approximation, covering a range of different applied scenarios to which our methods can be immediately applied. Throughout, we isolate and discuss special cases of our results which even allow for the construction of fast, and in particular linear, rates.

In this paper, we consider the mass-critical focusing nonlinear Schr\"odinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data sufficiently large can blow-up. More precisely, given a finite number of points, there exists particular profiles blowing up exactly at these points at the blow-up time. For pertubations of these profiles, we show that, with the help of an appropriate nonlinear feedback law located in an open set containing the blow-up points, the blow-up can be prevented from happening. More specifically, we construct a small-time control acting just before the blow-up time. The solution may then be extended globally in time. This is the first result of control for blow-up profiles for nonlinear Schr\"odinger type equations. Assuming further a geometrical control condition on the support of the control, we are able to prove a null-controllability result for such blow-up profiles. Finally, we discuss possible extensions to three-dimensional domains.

We present a modified simulated annealing method with a dynamical choice of the cooling temperature. The latter is determined via a closed-loop control and is proven to yield exponential decay of the entropy of the particle system. The analysis is carried out through kinetic equations for interacting particle systems describing the simulated annealing method in an extended phase space. Decay estimates are derived under the quasi-invariant scaling of the resulting system of Boltzmann-type equations to assess the consistency with their mean-field limit. Numerical results are provided to illustrate and support the theoretical findings.

This paper introduces a novel approach for cardinality-constrained Poisson regression to address feature selection challenges in high-dimensional count data. We formulate the problem as a mixed-integer conic optimization, enabling the use of modern solvers for optimal solutions. To enhance computational efficiency, we develop a safe screening based on Fenchel conjugates, thereby effectively removing irrelevant features before optimization. Experiments on synthetic datasets demonstrate that our safe screening significantly reduces the problem size, leading to substantial improvements in computational time. Our approach can solve Poisson regression problems with tens of thousands of features, exceeding the scale of previous studies. This work provides a valuable tool for interpretable feature selection in high-dimensional Poisson regression.

Proximal gradient methods are popular in sparse optimization as they are straightforward to implement. Nevertheless, they achieve biased solutions, requiring many iterations to converge. This work addresses these issues through a suitable feedback control of the algorithm's hyperparameter. Specifically, by designing an integral control that does not substantially impact the computational complexity, we can reach an unbiased solution in a reasonable number of iterations. In the paper, we develop and analyze the convergence of the proposed approach for strongly-convex problems. Moreover, numerical simulations validate and extend the theoretical results to the non-strongly convex framework.

The Open Loop Layout Problem (OLLP) seeks to position rectangular cells of varying dimensions on a plane without overlap, minimizing transportation costs computed as the flow-weighted sum of pairwise distances between cells. A key challenge in OLLP is to compute accurate inter-cell distances along feasible paths that avoid rectangle intersections. Existing approaches approximate inter-cell distances using centroids, a simplification that can ignore physical constraints, resulting in infeasible layouts or underestimated distances. This study proposes the first mathematical model that incorporates exact door-to-door distances and feasible paths under the Euclidean metric, with cell doors acting as pickup and delivery points. Feasible paths between doors must either follow rectangle edges as corridors or take direct, unobstructed routes. To address the NP-hardness of the problem, we present a metaheuristic framework with a novel encoding scheme that embeds exact path calculations. Experiments on standard benchmark instances confirm that our approach consistently outperforms existing methods, delivering superior solution quality and practical applicability.

This paper presents an intrinsic approach for addressing control problems with systems governed by linear ordinary differential equations (ODEs). We use computer algebra to constrain a Gaussian Process on solutions of ODEs. We obtain control functions via conditioning on datapoints. Our approach thereby connects Algebra, Functional Analysis, Machine Learning and Control theory. We discuss the optimality of the control functions generated by the posterior mean of the Gaussian Process. We present numerical examples which underline the practicability of our approach.

This paper proposes a Perturbed Proximal Gradient ADMM (PPG-ADMM) framework for solving general nonconvex composite optimization problems, where the objective function consists of a smooth nonconvex term and a nonsmooth weakly convex term for both primal variables. Unlike existing ADMM-based methods which necessitate the function associated with the last updated primal variable to be smooth, the proposed PPG-ADMM removes this restriction by introducing a perturbation mechanism, which also helps reduce oscillations in the primal-dual updates, thereby improving convergence stability. By employing a linearization technique for the smooth term and the proximal operator for the nonsmooth and weakly convex term, the subproblems have closed-form solutions, significantly reducing computational complexity. The convergence is established through a technically constructed Lyapunov function, which guarantees sufficient descent and has a well-defined lower bound. With properly chosen parameters, PPG-ADMM converges to an $\epsilon$-approximate stationary point at a sublinear convergence rate of $\mathcal{O}(1/\sqrt{K})$. Furthermore, by appropriately tuning the perturbation parameter $\beta$, it achieves an $\epsilon$-stationary point, providing stronger optimality guarantees. We further apply PPG-ADMM to two practical distributed nonconvex composite optimization problems, i.e., the distributed partial consensus problem and the resource allocation problem. The algorithm operates in a fully decentralized manner without a central coordinating node. Finally, numerical experiments validate the effectiveness of PPG-ADMM, demonstrating its improved convergence performance.

Decentralized Federated Learning (DFL) eliminates the reliance on the server-client architecture inherent in traditional federated learning, attracting significant research interest in recent years. Simultaneously, the objective functions in machine learning tasks are often nonconvex and frequently incorporate additional, potentially nonsmooth regularization terms to satisfy practical requirements, thereby forming nonconvex composite optimization problems. Employing DFL methods to solve such general optimization problems leads to the formulation of Decentralized Nonconvex Composite Federated Learning (DNCFL), a topic that remains largely underexplored. In this paper, we propose a novel DNCFL algorithm, termed \bf{DEPOSITUM}. Built upon proximal stochastic gradient tracking, DEPOSITUM mitigates the impact of data heterogeneity by enabling clients to approximate the global gradient. The introduction of momentums in the proximal gradient descent step, replacing tracking variables, reduces the variance introduced by stochastic gradients. Additionally, DEPOSITUM supports local updates of client variables, significantly reducing communication costs. Theoretical analysis demonstrates that DEPOSITUM achieves an expected $\epsilon$-stationary point with an iteration complexity of $\mathcal{O}(1/\epsilon^2)$. The proximal gradient, consensus errors, and gradient estimation errors decrease at a sublinear rate of $\mathcal{O}(1/T)$. With appropriate parameter selection, the algorithm achieves network-independent linear speedup without requiring mega-batch sampling. Finally, we apply DEPOSITUM to the training of neural networks on real-world datasets, systematically examining the influence of various hyperparameters on its performance. Comparisons with other federated composite optimization algorithms validate the effectiveness of the proposed method.

In this note we consider a problem of stochastic optimal control with the infinite-time horizon. We present analogues of the Seierstad sufficient conditions of overtaking optimality based on the dual variables stochastic described by BSDEs appeared in the Bismut-Pontryagin maximum principle.