Optimization and Control (math.OC)

Abstract: The classical Routh-Hurwitz criterion is one of the most popular methods to study the stability of polynomials with real coefficients, given its simplicity and ductility. However, when moving to polynomials with complex coefficients, a generalization exists but it is rather cumbersome and not as easy to apply. In this paper, we make such generalization clear and understandable for a wider public and develop an algorithm to apply it. After having explained the method, we demonstrate its use to determine the external stability of a system consisting of the interconnection between a rotating shaft and a PI-regulator. The extended Routh-Hurwitz criterion gives then necessary and sufficient conditions on the gains of the PI-regulator to achieve stabilization of the system together with regulation of the output. This illustrative example makes our formulation of the extended Routh-Hurwitz criterion ready to be used in several other applications.

Abstract: Orthogonal group synchronization is the problem of estimating $n$ elements $Z_1, \ldots, Z_n$ from the orthogonal group $\mathrm{O}(r)$ given some relative measurements $R_{ij} \approx Z_i^{}Z_j^{-1}$. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from $O(n)$ to $O(n^2)$. Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension $O(n^{3/2})$. For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that nonconvex relaxations of only slightly increased dimension seem sufficient for SLAM problems. We partially explain this. This also has implications for Kuramoto oscillators. Specifically, we minimize the least-squares cost function in terms of estimators $Y_1, \ldots, Y_n$. Each $Y_i$ is relaxed to the Stiefel manifold $\mathrm{St}(r, p)$ of $r \times p$ matrices with orthonormal rows. The available measurements implicitly define a (connected) graph $G$ on $n$ vertices. In the noiseless case, we show that second-order critical points are globally optimal as soon as $p \geq r+2$ for all connected graphs $G$. (This implies that Kuramoto oscillators on $\mathrm{St}(r, p)$ synchronize for all $p \geq r + 2$.) This result is the best possible for general graphs; the previous best known result requires $2p \geq 3(r + 1)$. For $p > r + 2$, our result is robust to modest amounts of noise (depending on $p$ and $G$). When local minima remain, they still achieve minimax-optimal error rates. Our proof uses a novel randomized choice of tangent direction to prove (near-)optimality of second-order critical points. Finally, we partially extend our noiseless landscape results to the complex case (unitary group), showing that there are no spurious local minima when $2p \geq 3r$.

Abstract: Stochastic Approximation has been a prominent set of tools for solving problems with noise and uncertainty. Increasingly, it becomes important to solve optimization problems wherein there is noise in both a set of constraints that a practitioner requires the system to adhere to, as well as the objective, which typically involves some empirical loss. We present the first stochastic approximation approach for solving this class of problems using the Ghost framework of incorporating penalty functions for analysis of a sequential convex programming approach together with a Monte Carlo estimator of nonlinear maps. We provide almost sure convergence guarantees and demonstrate the performance of the procedure on some representative examples.

Abstract: Drones are currently seen as a viable way for improving the distribution of parcels in urban and rural environments, while working in coordination with traditional vehicles like trucks. In this paper we consider the parallel drone scheduling vehicle routing problem, where the service of a set of customers requiring a delivery is split between a fleet of trucks and a fleet of drones. We consider two variations of the problem. In the first one the problem is more theoretical, and the target is the minimization of the time required to complete the service and have all the vehicles back to the depot. In the second variant more realistic constraints involving operating costs, capacity limitation and workload balance, are considered, and the target is to minimize the total operational costs. We propose several constraint programming models to deal with the two problems. An experimental champaign on the instances previously adopted in the literature is presented to validate the new solving methods. The results show that on top of being a viable way to solve problems to optimality, the models can also be used to derive effective heuristic solutions and high-quality lower bounds for the optimal cost, if the execution is interrupted after its natural end.

Abstract: We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\varepsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon\log(1/\varepsilon)+O(\varepsilon)$ for some explicit dimensional constants $C$ depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for $\mathcal{C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Abstract: In this paper, we study a continuous-time exploratory mean-variance (EMV) problem under the framework of reinforcement learning (RL), and the Choquet regularizers are used to measure the level of exploration. By applying the classical Bellman principle of optimality, the Hamilton-Jacobi-Bellman equation of the EMV problem is derived and solved explicitly via maximizing statically a mean-variance constrained Choquet regularizer. In particular, the optimal distributions form a location-scale family, whose shape depends on the choices of the Choquet regularizer. We further reformulate the continuous-time Choquet-regularized EMV problem using a variant of the Choquet regularizer. Several examples are given under specific Choquet regularizers that generate broadly used exploratory samplers such as exponential, uniform and Gaussian. Finally, we design a RL algorithm to simulate and compare results under the two different forms of regularizers.

Abstract: This note discusses an essentially decentralized interior point method, which is well suited for optimization problems arising in energy networks. Advantages of the proposed method are guaranteed and fast local convergence also for problems with non-convex constraints. Moreover, our method exhibits a small communication footprint and it achieves a comparably high solution accuracy with a limited number of iterations, whereby the local subproblems are of low computational complexity. We illustrate the performance of the proposed method on a problem from energy systems, i.e., we consider an optimal power flow problem with 708 buses.

Abstract: The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.

Abstract: We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart. Symmetric cones provide a unifying framework for some of the most important optimization models, including linear, second-order cone, and semidefinite optimization. Using tools from the field of Euclidean Jordan Algebras, we introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer. Using this structural result we show that SCMWU is a no-regret algorithm, and verify our theoretical results with extensive experiments. Our results unify and generalize the analysis for the Multiplicative Weights Update method over the probability simplex and the Matrix Multiplicative Weights Update method over the set of density matrices.

Abstract: In this paper, we consider a Markov decision process (MDP) with a Borel state space $\textbf{X}\cup\{\Delta\}$, where $\Delta$ is an absorbing state (cemetery), and a Borel action space $\textbf{A}$. We consider the space of finite occupation measures restricted on $\textbf{X}\times \textbf{A}$, and the extreme points in it. It is possible that some strategies have infinite occupation measures. Nevertheless, we prove that every finite extreme occupation measure is generated by a deterministic stationary strategy. Then, for this MDP, we consider a constrained problem with total undiscounted criteria and $J$ constraints, where the cost functions are nonnegative. By assumption, the strategies inducing infinite occupation measures are not optimal. Then, our second main result is that, under mild conditions, the solution to this constrained MDP is given by a mixture of no more than $J+1$ occupation measures generated by deterministic stationary strategies.

Abstract: We propose a new length formula that governs the iterates of the momentum method when minimizing differentiable semi-algebraic functions with locally Lipschitz gradients. It enables us to establish local convergence, global convergence, and convergence to local minimizers without assuming global Lipschitz continuity of the gradient, coercivity, and a global growth condition, as is done in the literature. As a result, we provide the first convergence guarantee of the momentum method starting from arbitrary initial points when applied to principal component analysis, matrix sensing, and linear neural networks.