Optimization and Control (math.OC)

Abstract: In this paper, for a function $f$ locally lower semicontinuous at a stationary point $\bar{x}$, we obtain complete characterizations of the Kurdyka-{\L}ojasiewicz (for short, K{\L}) property and the exact estimate of the K{\L} modulus via the outer limiting subdifferential of an auxilliary function, and obtain a sufficient condition for verifying sharpness of the K{\L} exponent. By introducing a $\frac{1}{1-\theta}$-th subderivative $h$ for $f$ at $\bar{x}$, we show that the K{\L} property of $f$ at $\bar{x}$ with exponent $\theta\in [0, 1)$ can be inherited by $h$ at $0$ with the same exponent $\theta$, and that the K{\L} modulus of $f$ at $\bar{x}$ is bounded above by that of $(1-\theta)h$ at $0$. When $\theta=\frac12$, we obtain the reverse results under the strong metrically subregularity of the subgradient mapping for the class of prox-regular, twice epi-differentiable and subdifferentially continuous functions by virtue of Moreau envelopes. We apply the obtained results to establish the K{\L} property with exponent $\frac12$ and to provide calculations of the K{\L} modulus for smooth functions, the pointwise max of finitely many smooth functions and the $\ell_p$ ($0<p\leq 1$) regularized functions respectively. It is worth noting that these functions often appear in structured optimization problems.

Abstract: Two central problems in modern control theory are the controller design problem: which deals with designing a control law for the dynamical system, and the state estimation problem (observer design problem): which deals with computing an estimate of the states of the dynamical system. The Linear Quadratic Regulator (LQR) and Kalman Filter (KF) solves these problems respectively for linear dynamical systems in an optimal manner, i.e., LQR is an optimal state feedback controller and KF is an optimal state estimator. In this note, we will be discussing the basic concepts, derivation, steady-state analysis, and numerical implementation of the LQR and KF.

Abstract: The paper develops a methodology for the design of coherent equalizing filters for quantum communication channels. Given a linear quantum system model of a quantum communication channel, the aim is to obtain another quantum system which, when coupled with the original system, mitigates degrading effects of the environment. The main result of the paper is a systematic equalizer synthesis algorithm which relies on methods of state-space robust control design via semidefinite programming.

Abstract: In this paper, we consider Riemannian online convex optimization with dynamic regret. First, we propose two novel algorithms, namely the Riemannian Online Optimistic Gradient Descent (R-OOGD) and the Riemannian Adaptive Online Optimistic Gradient Descent (R-AOOGD), which combine the advantages of classical optimistic algorithms with the rich geometric properties of Riemannian manifolds. We analyze the dynamic regrets of the R-OOGD and R-AOOGD in terms of regularity of the sequence of cost functions and comparators. Next, we apply the R-OOGD to Riemannian zero-sum games, leading to the Riemannian Optimistic Gradient Descent Ascent algorithm (R-OGDA). We analyze the average iterate and best-iterate of the R-OGDA in seeking Nash equilibrium for a two-player, zero-sum, g-convex-concave games. We also prove the last-iterate convergence of the R-OGDA for g-strongly convex-strongly concave problems. Our theoretical analysis shows that all proposed algorithms achieve results in regret and convergence that match their counterparts in Euclidean spaces. Finally, we conduct several experiments to verify our theoretical findings.

Abstract: We study the convergence of specific inexact alternating projections for two non-convex sets in a Euclidean space. The $\sigma$-quasioptimal metric projection ($\sigma \geq 1$) of a point $x$ onto a set $A$ consists of points in $A$ the distance to which is at most $\sigma$ times larger than the minimal distance $\mathrm{dist}(x,A)$. We prove that quasioptimal alternating projections, when one or both projections are quasioptimal, converge locally and linearly under the usual regularity assumptions on the two sets and their intersection. The theory is motivated by the successful application of alternating projections to low-rank matrix and tensor approximation. We focus on two problems -- nonnegative low-rank approximation and low-rank approximation in the maximum norm -- and develop fast alternating-projection algorithms for matrices and tensor trains based on cross approximation and acceleration techniques. The numerical experiments confirm that the proposed methods are efficient and suggest that they can be used to regularise various low-rank computational routines.

Abstract: Bus Rapid Transit (BRT) systems can provide a fast and reliable service to passengers at low investment costs compared to tram, metro and train systems. Therefore, they can be of great value to attract more passengers to use public transport. This paper thus focuses on the BRT investment problem: Which segments of a single bus line should be upgraded such that the number of newly attracted passengers is maximized? Motivated by the construction of a new BRT line around Copenhagen, we consider a setting in which multiple parties are responsible for different segments of the line. As each party has a limited willingness to invest, we solve a bi-objective problem to quantify the trade-off between the number of attracted passengers and the investment budget. We model different problem variations: First, we consider two potential passenger responses to upgrades on the line. Second, to prevent scattered upgrades along the line, we consider different restrictions on the number of upgraded connected components on the line. We propose an epsilon-constraint-based algorithm to enumerate the complete set of non-dominated points and investigate the complexity of this problem. Moreover, we perform extensive numerical experiments on artificial instances and a case study based on the BRT line around Copenhagen. Our results show that we can generate the full Pareto front for real-life instances and that the resulting trade-off between investment budget and attracted passengers depends both on the origin-destination demand and on the passenger response to upgrades. Moreover, we illustrate how the generated Pareto plots can assist decision makers in selecting from a set of geographical route alternatives in our case study.