Optimization and Control (math.OC)

Abstract: We propose a novel approach to certify closed-loop stability and safety of a constrained polynomial system based on the combination of Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs). For polynomial systems that are affine in the control input, both classes of functions can be constructed via Sum Of Squares (SOS) programming. Using two versions of the Positivstellensatz we derive an SOS formulation seeking a rational controller that - if feasible - results in compatible CLF and multiple CBFs.

Abstract: This paper studies a structure-preserving model reduction problem for large-scale second-order dynamical systems via the framework of time-domain moment matching. The moments of a second-order system are interpreted as the solutions of second-order Sylvester equations, which leads to families of parameterized second-order reduced models that match the moments of an original second-order system at selected interpolation points. Based on this, a two-sided moment matching problem is addressed, providing a unique second-order reduced system that match two distinct sets interpolation points. Furthermore, we also construct the reduced second-order systems that matches the moments of both zero and first order derivative of the original second-order system. Finally, the Loewner framework is extended to the second-order systems, where two parameterized families of models are presented that retain the second-order structure and interpolate sets of tangential data.

Abstract: High demand, rising customer expectations, and government regulations are forcing companies to increase the efficiency and sustainability of urban (last-mile) distribution. Consequently, several new delivery concepts have been proposed that increase flexibility for customers and other stakeholders. One of these innovations is crowdsourced delivery, where deliveries are made by occasional drivers who wish to utilize their surplus resources (unused transport capacity) by making deliveries in exchange for some compensation. In addition to reducing delivery costs, the potential benefits of crowdsourced delivery include better utilization of transport capacity, a reduction in overall traffic, and increased flexibility (by scaling up and down delivery capacity as needed). The use of occasional drivers poses new challenges because (unlike traditional couriers) neither their availability nor their behavior in accepting delivery offers is certain. The relationship between the compensation offered to occasional drivers and the probability that they will accept a task has been largely neglected in the scientific literature. Therefore, we consider a setting in which compensation-dependent acceptance probabilities are explicitly considered in the process of assigning delivery tasks to occasional drivers. We propose a mixed-integer nonlinear model that minimizes the expected delivery costs while identifying optimal assignments of tasks to a mix of traditional and occasional drivers and their compensation. We propose exact linearization schemes for two practically relevant probability functions and an approximate linearization scheme for the general case. The results of our computational study show clear advantages of our new approach over existing ones.

Abstract: This paper develops projection-free algorithms for online convex optimization with stochastic constraints. We design an online primal-dual projection-free framework that can take any projection-free algorithms developed for online convex optimization with no long-term constraint. With this general template, we deduce sublinear regret and constraint violation bounds for various settings. Moreover, for the case where the loss and constraint functions are smooth, we develop a primal-dual conditional gradient method that achieves $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations. Furthermore, for the setting where the loss and constraint functions are stochastic and strong duality holds for the associated offline stochastic optimization problem, we prove that the constraint violation can be reduced to have the same asymptotic growth as the regret.

Abstract: While gradient based methods are ubiquitous in machine learning, selecting the right step size often requires "hyperparameter tuning". This is because backtracking procedures like Armijo's rule depend on quality evaluations in every step, which are not available in a stochastic context. Since optimization schemes can be motivated using Taylor approximations, we replace the Taylor approximation with the conditional expectation (the best $L^2$ estimator) and propose "Random Function Descent" (RFD). Under light assumptions common in Bayesian optimization, we prove that RFD is identical to gradient descent, but with calculable step sizes, even in a stochastic context. We beat untuned Adam in synthetic benchmarks. To close the performance gap to tuned Adam, we propose a heuristic extension competitive with tuned Adam.

Abstract: With the increasing complexity of power systems, faster methods for power system reliability analysis are needed. We propose a novel methodology to solve the security constrained optimal power flow (SCOPF) problem that reduces the computational time by using the Sherman-Morrison-Woodbury identity and Benders decomposition. The case study suggests that in a 500 node system, the run time is reduced by 83.5% while ensuring a reliable operation of the system considering short- and long-term post-contingency limits and reducing the operational costs, compared to a preventive `N-1' strategy.

Abstract: We consider a deterministic mean field games problem in which a typical agent solves an optimal control problem where the dynamics is affine with respect to the control and the cost functional has a growth which is polynomial with respect to the state variable. In this framework, we construct a mean field game problem in discrete time and finite state space that approximates equilibria of the original game. Two numerical examples, solved with the fictitious play method, are presented.

Abstract: In this paper we formulate and solve an optimal problem for Stochastic process with a regime absorbing state. The solution for this problem is obtained through a system of partial differential equations. The method is applied to obtain an explicit solution for the Merton portfolio problem when an asset has a default probability in case of a log utility.