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This paper studies periodic event-triggered networked control for nonlinear systems, where the plants and controllers are connected by multiple independent communication channels. Several network-induced imperfections are considered simultaneously, including time-varying inter-sampling times, sensor node scheduling, and especially, large time-varying transmission delays, where the transmitted signal may arrive at the destination node after the next transmission occurs. A new hybrid system approach is provided to model the closed-loop system that contains all communication related behavior. Then, by constructing new storage functions on the system state and updating errors, the relationship between the maximum allowable sampling period (MASP) and maximum allowable delay number in sampling (MADNS) is analyzed, where the latter denotes how many inter-sampling periods can be included in one transmission delay. Moreover, to efficiently reduce unnecessary transmissions, a new dynamic event-triggered control scheme is proposed, where the event-triggering conditions are detected only at aperiodic and asynchronous sampling instants. From emulation-based method, where the controllers are initially designed by ignoring all the network-induced imperfections, sufficient conditions on the dynamic event-triggered control are given to ensure closed-loop input-to-state stability with respect to external disturbances. Moreover, according to different capacities of the communication channels, the corresponding implementation strategies of the designed dynamic event-triggered control schemes are discussed. Finally, two nonlinear examples are simulated to illustrate the feasibility and efficiency of the theoretical results., Comment: 18 pages, 6 figures, 3 tables

Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS) while at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows for use of arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper recursive $O\left( n\right) $ Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. The MBS motion equations are derived in closed form using the Lie group formulation. One are the so-called 'Euler-Jourdain' or 'projection' equations, of which Kane's equations are a special case, and the other are the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

We consider a distributed server system consisting of a large number of servers, each with limited capacity on multiple resources (CPU, memory, disk, etc.). Jobs with different rewards arrive over time and require certain amounts of resources for the duration of their service. When a job arrives, the system must decide whether to admit it or reject it, and if admitted, in which server to schedule the job. The objective is to maximize the expected total reward received by the system. This problem is motivated by control of cloud computing clusters, in which, jobs are requests for Virtual Machines or Containers that reserve resources for various services, and rewards represent service priority of requests or price paid per time unit of service by clients. We study this problem in an asymptotic regime where the number of servers and jobs' arrival rates scale by a factor L, as L becomes large. We propose a resource reservation policy that asymptotically achieves at least 1/2, and under certain monotone property on jobs' rewards and resources, at least 11/4 of the optimal expected reward. The policy automatically scales the number of VM slots for each job type as the demand changes, and decides in which servers the slots should be created in advance, without the knowledge of traffic rates. It effectively tracks a low-complexity greedy packing of existing jobs in the system while maintaining only a small number, g(L) = w(logL), of reserved VM slots for high priority jobs that pack well.

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints., arXiv admin note: substantial text overlap with arXiv:1806.06230

Following the breakthrough work of Tardos (Oper Res 34:250–256, 1986) in the bit-complexity model, Vavasis and Ye (Math Program 74(1):79–120, 1996) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $$\max \, c^\top x,\, Ax = b,\, x \ge 0,\, A \in \mathbb {R}^{m \times n}$$ max c ⊤ x , A x = b , x ≥ 0 , A ∈ R m × n , Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that $$O(n^{3.5} \log (\bar{\chi }_A+n))$$ O ( n 3.5 log ( χ ¯ A + n ) ) iterations suffice to solve (LP) exactly, where $$\bar{\chi }_A$$ χ ¯ A is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J Optim 13(4):1054–1079, 2003), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure $$\bar{\chi }^*_A$$ χ ¯ A ∗ , defined as the minimum $$\bar{\chi }_{AD}$$ χ ¯ AD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $$O(m^2 n^2 + n^3)$$ O ( m 2 n 2 + n 3 ) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying $$\bar{\chi }_{AD} \le n(\bar{\chi }_A^*)^3$$ χ ¯ AD ≤ n ( χ ¯ A ∗ ) 3 . This algorithm also allows us to approximate the value of $$\bar{\chi }_A$$ χ ¯ A up to a factor $$n (\bar{\chi }_A^*)^2$$ n ( χ ¯ A ∗ ) 2 . This result is in surprising contrast to that of Tunçel (Math Program 86(1):219–223, 1999), who showed NP-hardness for approximating $$\bar{\chi }_A$$ χ ¯ A to within $$2^{\textrm{poly}(\textrm{rank}(A))}$$ 2 poly ( rank ( A ) ) . The key insight for our algorithm is to work with ratios $$g_i/g_j$$ g i / g j of circuits of A—i.e., minimal linear dependencies $$Ag=0$$ A g = 0 —which allow us to approximate the value of $$\bar{\chi }_A^*$$ χ ¯ A ∗ by a maximum geometric mean cycle computation in what we call the ‘circuit ratio digraph’ of A. While this resolves Monteiro and Tsuchiya’s question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $$O(n^{2.5} \log (n)\log (\bar{\chi }^*_A+n))$$ O ( n 2.5 log ( n ) log ( χ ¯ A ∗ + n ) ) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $$n/\log n$$ n / log n improvement on the iteration complexity bound of the original Vavasis–Ye algorithm.

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, where the numerator can be written as the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous (possibly nonconvex) function, and the denominator is weakly convex over the constraint set. This model problem includes the composite optimization problems studied extensively lately, and encompasses many important modern fractional optimization problems arising from diverse areas such as the recently proposed scale invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and any of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted Fast Iterative Shrinking-Threshold Algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka--{\L}ojasiewicz (KL) property. In particular, our algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. In the case where the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose an enhanced extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples., Comment: Revised version: Oct. 16, 2020

This paper is about obtaining stable vehicle platooning by using Cooperative Adaptive Cruise Control when the communication is unreliable and suffers from message losses. We model communication losses as independent random events and we propose an original design for the cooperative controller, which mitigates the effect of losses. This objective is obtained by a switching controller that has a twofold objective: on the one hand, it promotes both plant stability and string stability of the average error dynamics by an $H_infty$ approach, and on the other hand it minimizes the variance around the average. We show by simulations that the proposed controller is able to compensate even for high probability of losses., Comment: 10 pages, 7 figures; submitted to journal

Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A key difference between renewables and traditional generators is that many renewable resources are managed by individuals, especially in the distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producers compete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, even when there are a large number of similar players. In contrast, we show that due to the inherent randomness of renewable resources, the equilibria in our capacity game becomes efficient as the number of players grows and coincides with the centralized decision from the social planner's problem. This result provides a new perspective on how to look at the positive influence of randomness in a game framework as well as its contribution to resource planning, scheduling, and bidding. We validate our results by simulation studies using real world data., Appears in IEEE Transactions on Sustainable Computing

We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set, and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual offline setting, we present SampleGreedy, which obtains a $(p + 2 + o(1))$-approximation for maximizing a submodular function subject to a $p$-extendible system using $O(n + nk/p)$ evaluation and feasibility queries, where $k$ is the size of the largest feasible set. The approximation ratio improves to $p+1$ and $p$ for monotone submodular and linear objectives, respectively. In the streaming setting, we present SampleStreaming, which obtains a $(4p +2 - o(1))$-approximation for maximizing a submodular function subject to a $p$-matchoid using $O(k)$ memory and $O(km/p)$ evaluation and feasibility queries per element, where $m$ is the number of matroids defining the $p$-matchoid. The approximation ratio improves to $4p$ for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks., Comment: arXiv admin note: text overlap with arXiv:1802.07098

This paper addresses the single-item single-stocking location non-stationary stochastic lot-sizing problem under a reorder point – order quantity control strategy. The reorder points and order quantities are chosen at the beginning of the planning horizon. The reorder points are allowed to vary with time and we consider order quantities either to be a series of time-dependent constants or a fixed value; this leads to two variants of the policy: the ( s t , Q t ) and the ( s t , Q ) policies, respectively. For both policies, we present stochastic dynamic programs (SDP) to determine optimal policy parameters and introduce mixed integer non-linear programming (MINLP) heuristics that leverage piecewise-linear approximations of the cost function. Numerical experiments demonstrate that our solution method efficiently computes near-optimal parameters for a broad class of problem instances.

This paper deals with the simultaneous estimation of the attitude, position and linear velocity for vision-aided inertial navigation systems. We propose a nonlinear observer on $SO(3)\times \mathbb{R}^{15}$ relying on body-frame acceleration, angular velocity and (stereo or monocular) bearing measurements of some landmarks that are constant and known in the inertial frame. Unlike the existing local Kalman-type observers, our proposed nonlinear observer guarantees almost global asymptotic stability and local exponential stability. A detailed uniform observability analysis has been conducted and sufficient conditions are derived. Moreover, a hybrid version of the proposed observer is provided to handle the intermittent nature of the measurements in practical applications. Simulation and experimental results are provided to illustrate the effectiveness of the proposed state observer., Accepted for publication in IEEE Transaction on Automatic Control. 16 pages, 6 figures

Optimal sensor and actuator selection is a central challenge in high-dimensional estimation and control. Nearly all subsequent control decisions are affected by these sensor/actuator locations, and optimal placement amounts to an intractable brute-force search among the combinatorial possibilities. In this work, we exploit balanced model reduction and greedy optimization to efficiently determine sensor and actuator selections that optimize observability and controllability. In particular, we determine locations that optimize scalar measures of observability and controllability via greedy matrix QR pivoting on the dominant modes of the direct and adjoint balancing transformations. Pivoting runtime scales linearly with the state dimension, making this method tractable for high-dimensional systems. The results are demonstrated on the linearized Ginzburg-Landau system, for which our algorithm approximates known optimal placements computed using costly gradient descent methods., Comment: 8 pages, 6 figures

We study a noisy tensor completion problem of broad practical interest, namely, the reconstruction of a low-rank tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on "incoherent" and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm -- (vanilla) gradient descent following a rough initialization -- that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all individual tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e. minimal sample complexity and optimal estimation accuracy). The estimation errors are evenly spread out across all entries, thus achieving optimal $\ell_{\infty}$ statistical accuracy. We have also discussed how to extend our approach to accommodate asymmetric tensors. The insight conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems., Comment: Accepted to Operations Research

In this paper, we investigate the relationships between proper efficiency and the solutions of a general scalarization problem in multi-objective optimization. We provide some conditions under which the solutions of the dealt with scalar program are properly efficient and vice versa. We also show that, under some conditions, if the considered general scalar problem is unbounded, then the original multi-objective problem does not have any properly efficient solution. In another part of the work, we investigate a general transformation of the objective functions which preserves proper efficiency. We show that several important results existing in the literature are direct consequences of the results of the present paper.

Identifying the multiple critical components in power systems whose absence together has severe impact on system performance is a crucial problem for power systems known as $(N-x)$ contingency analysis. However, the inherent combinatorial feature of the $N-x$ contingency analysis problem incurs by the increase of $x$ in the $(N-x)$ term, making the problem intractable for even relatively small test systems. We present a new framework for identifying the $N-x$ contingencies that captures both topology and physics of the network. Graph theory provides many ways to measure power grid graphs, i.e., buses as nodes and lines as edges, allowing researchers to characterize system structure and optimize algorithms. This article proposes a scalable approach based on the group betweenness centrality concept that measures the impact of multiple components in the electric power grid as well as line outage distribution factors that find the lines whose loss has the highest impact on the power flow in the network. The proposed approach is a quick and efficient solution for identifying the most critical lines in power networks. The proposed approach is validated using various test cases, and results show that the proposed approach is able to quickly identify multiple contingencies that result in violations.

The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations. Our first formulation is an improvement of a previous formulation. It significantly decreases the number of constraints while preserving the optimal value of the linear relaxation. Our second formulation contains less variables and constraints but it has a weaker linear relaxation bound. We besides introduce an algorithm which enables us to compute strong bounds and significantly reduce the size of our formulations. Finally, the efficiency of the algorithm and the proposed formulations are compared in terms of quality of the linear relaxation and computation time over instances from OR-Library., Lecture Notes in Computer Science 2018

We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: 1) a single-layer, fixed-step algorithm with convergence guarantee for general (noncocoercive, nonstrictly) monotone aggregative games and 2) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of the algorithms via (alternating) inertial and over-relaxation steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary choosing a prior—that is, a pair of signal and noise distributions ranging over independent Wasserstein balls—with the goal to minimize and maximize the expected squared estimation error, respectively. We show that, if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, by which the players’ optimal strategies are given by an affine estimator and a normal prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank–Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general-purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form. Funding: This research was supported by the Swiss National Science Foundation [Grants BSCGI0_ 157733 and 51NF40_180545], an Early Postdoc.Mobility Fellowship [Grant P2ELP2_195149], and the European Research Council [Grant TRUST-949796].

Following the risk-taking model of Seel and Strack, $n$ players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the problem of a principal who aims to induce a desirable equilibrium performance of the players by choosing how much reward is attributed to each rank. Specifically, we determine optimal reward schemes for principals interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape., To appear in SIAM Journal on Financial Mathematics

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset’s elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation.

The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way, we develop extended calculus rules for first-order and second-order generalized differential constructions while paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers, strong metric subregularity of Karush-Kuhn-Tucker systems in parametric optimization, and so on.

This work focuses on finding optimal locations for charging stations for one-way electric car sharing programs. The relocation of vehicles by a service staff is generally required in vehicle sharing programs in order to correct imbalances in the network. We seek to limit the need for vehicle relocation by strategically locating charging stations given estimates of traffic flow. A mixed-integer linear programming formulation is presented with a large number of potential charging station locations. A column generation approach is used which finds an optimal set of locations for the continuous relaxation of our problem. Results of a numerical experiment using real traffic and geographic information system location data show that our formulation significantly increases the balanced flow across the network, while our column generation technique was found to produce a superior solution in much shorter computation time compared to solving the original formulation with all possible station locations.

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multi-term disjunction by considering the convex set's recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set., 25 pages, 11 figures

This work considers the problem of resilient consensus, where stochastic values of trust between agents are available. Specifically, we derive a unified mathematical framework to characterize convergence, deviation of the consensus from the true consensus value, and expected convergence rate, when there exists additional information of trust between agents. We show that under certain conditions on the stochastic trust values and consensus protocol: First, almost sure convergence to a common limit value is possible even when malicious agents constitute more than half of the network connectivity; second, the deviation of the converged limit, from the case where there is no attack, i.e., the true consensus value, can be bounded with probability that approaches 1 exponentially; and third correct classification of malicious and legitimate agents can be attained in finite time almost surely. Furthermore, the expected convergence rate decays exponentially as a function of the quality of the trust observations between agents.

In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of models which achieve moment matching. However, differently from the deterministic case, these reduced-order models cannot be considered "simpler" because of the high computational cost paid to determine the moment. To overcome this difficulty, we relax the moment matching problem in two different ways and we present two classes of reduced-order models which, approximately matching the stochastic moment, are computationally tractable., This article has been accepted for publication by IEEE Transactions on Automatic Control. The manuscript included in this file is the open access accepted version. This open access version is released on arXiv in accordance with the IEEE copyright agreement

In a Hilbert space, we provide a fast dynamic approach to the hierarchical minimization problem which consists in finding the minimum norm solution of a convex minimization problem. For this, we study the convergence properties of the trajectories generated by a damped inertial dynamic with Tikhonov regularization. When the time goes to infinity, the Tikhonov regularization parameter is supposed to tend towards zero, not too fast, which is a key property to make the trajectories strongly converge towards the minimizer of $f$ of minimum norm. According to the structure of the heavy ball method for strongly convex functions, the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Therefore, it also converges to zero, which will ensure rapid convergence of values. Precisely, under a proper tuning of these parameters, based on Lyapunov's analysis, we show that the trajectories strongly converge towards the minimizer of minimum norm, and we provide the convergence rate of the values. We show a trade off between the property of fast convergence of values, and the property of strong convergence towards the minimum norm solution. This study improves several previous works where this type of results was obtained under restrictive hypotheses., 22 pages, 1 figure. arXiv admin note: text overlap with arXiv:2104.11987

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth., 37 pages, 6 figures, 5 tables To Appear in Computational Optimization and Applications

International audience; This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by applications to some models in behavioral sciences, we establish new variational principles that can be viewed as far-going extensions of the Ekeland variational principle to cover domination vector settings. Our approach combines advantages of both primal and dual techniques in variational analysis with providing useful sufficient conditions for the existence of variational traps in behavioral science models with variable domination structures.

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For N P -hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing.

Problem definition: We consider an intermediary’s problem of dynamically matching demand and supply of heterogeneous types in a periodic-review fashion. Specifically, there are two disjoint sets of demand and supply types, and a reward for each possible matching of a demand type and a supply type. In each period, demand and supply of various types arrive in random quantities. The platform decides on the optimal matching policy to maximize the expected total discounted rewards, given that unmatched demand and supply may incur waiting or holding costs, and will be fully or partially carried over to the next period. Academic/practical relevance: The problem is crucial to many intermediaries who manage matchings centrally in a sharing economy. Methodology: We formulate the problem as a dynamic program. We explore the structural properties of the optimal policy and propose heuristic policies. Results: We provide sufficient conditions on matching rewards such that the optimal matching policy follows a priority hierarchy among possible matching pairs. We show that those conditions are satisfied by vertically and unidirectionally horizontally differentiated types, for which quality and distance determine priority, respectively. Managerial implications: The priority property simplifies the matching decision within a period, and the trade-off reduces to a choice between matching in the current period and that in the future. Then the optimal matching policy has a match-down-to structure when considering a specific pair of demand and supply types in the priority hierarchy.

This paper introduces ROmodel, an open source Python package extending the modeling capabilities of the algebraic modeling language Pyomo to robust optimization problems. ROmodel helps practitioners transition from deterministic to robust optimization through modeling objects which allow formulating robust models in close analogy to their mathematical formulation. ROmodel contains a library of commonly used uncertainty sets which can be generated using their matrix representations, but it also allows users to define custom uncertainty sets using Pyomo constraints. ROmodel supports adjustable variables via linear decision rules. The resulting models can be solved using ROmodels solvers which implement both the robust reformulation and cutting plane approach. ROmodel is a platform to implement and compare custom uncertainty sets and reformulations. We demonstrate ROmodel’s capabilities by applying it to six case studies. We implement custom uncertainty sets based on (warped) Gaussian processes to show how ROmodel can integrate data-driven models with optimization.

Multistage stochastic programming provides a modeling framework for sequential decision-making problems that involve uncertainty. One typically overlooked aspect of this methodology is how uncertainty is incorporated into modeling. Traditionally, statistical forecasting techniques with simple forms, e.g., (first-order) autoregressive time-series models, are used to extract scenarios to be added to optimization models to represent the uncertain future. However, often times, the performance of these forecasting models are not thoroughly assessed. Motivated by the advances in probabilistic forecasting, we incorporate a deep learning-based time-series forecasting method into multistage stochastic programming framework, and compare it with the cases where a traditional forecasting method is employed to model the uncertainty. We assess the impact of more accurate forecasts on the quality of two commonly used look-ahead policies, a deterministic one and a two-stage one, in a rolling\red{-}horizon framework on a practical problem. Our results illustrate that more accurate forecasts contribute substantially to the model performance, and enable obtaining high-quality solutions even from computationally cheap heuristics. They also show that the probabilistic forecasting capabilities of deep learning-based methods can be especially beneficial when used as a (conditional) sampling tool for scenario-based models, and to predict the worst-case scenario for risk-averse models.

We study the non-convex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This model is motivated by applications in molecular microscopy and image processing, where each measurement of an unknown object is subject to an independent random rotation from a rotational group. Equivalently, it is a Gaussian mixture model where the mixture centers belong to a group orbit. We show that fundamental properties of the likelihood landscape depend on the signal-to-noise ratio and the group structure. At low noise, this landscape is "benign" for any discrete group, possessing no spurious local optima and only strict saddle points. At high noise, this landscape may develop spurious local optima, depending on the specific group. We discuss several positive and negative examples, and provide a general condition that ensures a globally benign landscape. For cyclic permutations of coordinates on $\mathbb{R}^d$ (multi-reference alignment), there may be spurious local optima when $d \geq 6$, and we establish a correspondence between these local optima and those of a surrogate function of the phase variables in the Fourier domain. We show that the Fisher information matrix transitions from resembling that of a single Gaussian in low noise to having a graded eigenvalue structure in high noise, which is determined by the graded algebra of invariant polynomials under the group action. In a local neighborhood of the true object, the likelihood landscape is strongly convex in a reparametrized system of variables given by a transcendence basis of this polynomial algebra. We discuss implications for optimization algorithms, including slow convergence of expectation-maximization, and possible advantages of momentum-based acceleration and variable reparametrization for first- and second-order descent methods.

Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and numerical perspectives. Namely, we show convergence of the regularization method and establish convergence properties of a couple of majorization approaches for the associated nonconvex problems. We also test a monotone algorithm for an academic example where the operator is an $M$ matrix, and on a time-dependent optimal control problem, pointing out the advantages of employing variable penalties over a fixed penalty.

Data centers are increasingly using high-speed circuit switches to cope with the growing demand and reduce operational costs. One of the fundamental tasks of circuit switches is to compute a sparse collection of switching configurations to support a traffic demand matrix. Such a problem has been addressed in the literature with variations of the approach proposed by Birkhoff in 1946 to decompose a doubly stochastic matrix exactly. However, the existing methods are heuristic and do not have theoretical guarantees on how well a collection of switching configurations (i.e., permutations) can approximate a traffic matrix (i.e., a scaled doubly stochastic matrix). In this paper, we revisit Birkhoff’s approach and make three contributions. First, we establish the first theoretical bound on the sparsity of Birkhoff’s algorithm (i.e., the number of switching configurations necessary to approximate a traffic matrix). In particular, we show that by using a subset of the admissible permutation matrices, Birkhoff’s algorithm obtains an ǫ-approximate decomposition with at most O(log(1/ǫ)) permutations. Second, we propose a new algorithm, Birkhoff+, which combines the wealth of Frank-Wolfe with Birkhoff’s approach to obtain sparse decompositions in a fast manner. And third, we evaluate the performance of the proposed algorithm numerically and study how this affects the performance of a circuit switch. Our results show that Birkhoff+ is superior to previous algorithms in terms of throughput, running time, and number of switching configurations.

Real-time, guaranteed safe trajectory planning is vital for navigation in unknown environments. However, real-time navigation algorithms typically sacrifice robustness for computation speed. Alternatively, provably safe trajectory planning tends to be too computationally intensive for real-time replanning. We propose FaSTrack, Fast and Safe Tracking, a framework that achieves both real-time replanning and guaranteed safety. In this framework, real-time computation is achieved by allowing any trajectory planner to use a simplified \textit{planning model} of the system. The plan is tracked by the system, represented by a more realistic, higher-dimensional \textit{tracking model}. We precompute the tracking error bound (TEB) due to mismatch between the two models and due to external disturbances. We also obtain the corresponding tracking controller used to stay within the TEB. The precomputation does not require prior knowledge of the environment. We demonstrate FaSTrack using Hamilton-Jacobi reachability for precomputation and three different real-time trajectory planners with three different tracking-planning model pairs., Published in the IEEE Transactions on Automatic Control

The problem of synthesizing an optimal sensor selection policy is pertinent to a variety of engineering applications ranging from event detection to autonomous navigation. We consider such a synthesis problem over an infinite time horizon with a discounted cost criterion. We formulate this problem in terms of a value iteration over the continuous space of covariance matrices. To obtain a computationally tractable solution, we subsequently formulate an approximate sensor selection problem, which is solvable through a point-based value iteration over a finite "mesh" of covariance matrices with a user-defined bounded trace. We provide theoretical guarantees bounding the suboptimality of the sensor selection policies synthesized through this method and provide numerical examples comparing them to known results., Comment: 9 pages, 2 figures

In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications.In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

As electric vehicle (EV) technologies become mature, EV has been rapidly adopted in modern transportation systems, and is expected to provide future autonomous mobility-on-demand (AMoD) service with economic and societal benefits. However, EVs require frequent recharges due to their limited and unpredictable cruising ranges, and they have to be managed efficiently given the dynamic charging process. It is urgent and challenging to investigate a computationally efficient algorithm that provide EV AMoD system performance guarantees under model uncertainties, instead of using heuristic demand or charging models. To accomplish this goal, this work designs a data-driven distributionally robust optimization approach for vehicle supply-demand ratio and charging station utilization balancing, while minimizing the worst-case expected cost considering both passenger mobility demand uncertainties and EV supply uncertainties. We then derive an equivalent computationally tractable form for solving the distributionally robust problem in a computationally efficient way under ellipsoid uncertainty sets constructed from data. Based on E-taxi system data of Shenzhen city, we show that the average total balancing cost is reduced by 14.49%, the average unfairness of supply-demand ratio and utilization is reduced by 15.78% and 34.51% respectively with the distributionally robust vehicle balancing method, compared with solutions which do not consider model uncertainties., This paper has been published in IROS2020

We show how to value both variable generation and energy storage to enable them to be integrated fairly and optimally into electricity capacity markets. We develop theory based on balancing expected energy unserved against costs of capacity procurement, and in which the optimal procurement is that necessary to meet an appropriate reliability standard. For conventional generation the theory reduces to that already in common use. Further the valuation of both variable generation and storage coincides with the traditional risk-based approach based on equivalent firm capacity. The determination of the equivalent firm capacity of storage requires particular care; this is due both to the flexibility with which storage added to an existing system may be scheduled, and also because, when any resource is added to an existing system, storage already within that system may be flexibly rescheduled. We illustrate the theory with an example based on the GB system., To appear in The Energy Journal

The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history that dates back to work by Cheney and Goldstein in 1959. In 2018, Aharoni, Censor, and Jiang revisited this problem and proposed an algorithm that can be used when the two sets are finite intersections of halfspaces. Motivated by their work, we present alternative algorithms that utilize projection and proximity operators. Our modeling framework is able to accommodate even convex sets. Numerical experiments indicate that these methods are competitive and sometimes superior to the one proposed by Aharoni et al.

Stability of retarded differential equations is closely related to the existence of Lyapunov-Krasovskii functionals. Even if a number of converse results have been reported regarding the existence of such functionals, there is a lack of constructive methods for their selection. For certain classes of time-delay systems for which such constructive methods are lacking, it was shown that Lyapunov-Krasovskii functionals that are also allowed to depend on the time-derivative of the state-trajectory are efficient tools for the study of the stability properties. However, in such an approach the initial condition needs to be assumed absolutely continuous with a square integrable weak derivative. In addition, the stability results hold for initial conditions that are evaluated based on the magnitude of both the initial condition and its time-derivative. The main objective of this paper is to show that, for certain classes of time-delay systems, the aforementioned stability results can actually be extended to initial conditions that are only assumed continuous and that are evaluated in uniform norm. European Commission - European Regional Development Fund Science Foundation Ireland

This paper conducts a comparative study of proximal gradient methods (PGMs) and proximal DC algorithms (PDCAs) for sparse regression problems which can be cast as Difference-of-two-Convex-functions (DC) optimization problems. It has been shown that for DC optimization problems, both General Iterative Shrinkage and Thresholding algorithm (GIST), a modified version of PGM, and PDCA converge to critical points. Recently some enhanced versions of PDCAs are shown to converge to d-stationary points, which are stronger necessary condition for local optimality than critical points. In this paper we claim that without any modification, PGMs converge to a d-stationary point not only to DC problems but also to more general nonsmooth nonconvex problems under some technical assumptions. While the convergence to d-stationary points is known for the case where the step size is small enough, the finding of this paper is valid also for extended versions such as GIST and its alternating optimization version, which is to be developed in this paper. Numerical results show that among several algorithms in the two categories, modified versions of PGM perform best among those not only in solution quality but also in computation time., Theorem 5.2. and its proof; We have added an assumption and modified the proof because the previous version (and Theorem 2 of the published version in Optimization Letters) had some mathematical errors. Furthermore, we have corrected some typos

Metaheuristics are widely employed to solve hard optimization problems, like vehicle routing problems (VRP), for which exact solution methods are impractical. In particular, local search-based metaheuristics have been successfully applied to the capacitated VRP (CVRP). The CVRP aims at defining the minimum-cost delivery routes for a given set of identical vehicles since each vehicle only travels one route and there is a single (central) depot. The best metaheuristics to the CVRP avoid getting stuck in local optima by embedding specific hill-climbing mechanisms such as diversification strategies into the solution methods. This paper introduces a hybridization of a novel adaptive version of Iterated Local Search with Path-Relinking (AILS-PR) to the CVRP. The major contribution of this paper is an automatic mechanism to control the diversity step of the metaheuristic to allow it to escape from local optima. The results of experiments with 100 benchmark CVPR instances show that AILS-PR outperformed the state-of-the-art CVRP metaheuristics.

The motivation for this work stems from the problem of scheduling requests for flow at supply points located throughout an automated network of open-water channels. The off-take flows are rigid-profile inputs to the system dynamics. In particular, the channel operator can only shift orders in time to satisfy constraints on the automatic response to changes in the load. This leads to a nonconvex semi-infinite programming problem, with a sum-separable cost that encodes the collective sensitivity of end-users to scheduling delays. The constraints encode the linear time-invariant continuous-time dynamics and limits on the state across a continuous scheduling horizon. Discretization is used to arrive at a more manageable approximation of the semi-infinite program. A method for parsimoniously refining the discretization is applied to ensure continuous-time feasibility for solutions of the approximate problem. It is then shown how to improve the cost without loss of feasibility. Supporting analysis is provided, along with simulation results for a realistic irrigation channel setup to illustrate the approach.

Mobile medical units (MMUs) are customized vehicles fitted with medical equipment that are used to provide primary care in rural environments. As MMUs can be easily relocated, they enable a demand-oriented, flexible, and local provision of health services. In this paper, we investigate the strategic planning of an MMU service by deciding where MMU operation sites should be set up and how often these should be serviced. To that end, we study the strategic planning problem for MMUs ( SPMMU ) – a capacitated set covering problem that includes existing practices and two types of patient demands: (i) steerable demands representing patients who seek health services through a centralized appointment system and can be steered to any treatment facility within a given consideration set and (ii) unsteerable demands representing walk-in patients who always visit the closest available treatment facility. We propose an integer linear program for the SPMMU that can be solved via Benders decomposition and constraint generation. Starting from this formulation, we focus on the uncertain version of the problem in which steerable and unsteerable demands are modeled as random variables that may vary within a given interval. Using methods from robust optimization and duality theory, we devise exact constraint generation methods to solve the robust counterparts for interval and budgeted uncertainty sets. All our results transfer to the session-specific SPMMU and we evaluate our models in a computational study based on a set of instances generated from a rural primary care system in Germany.

We propose a new framework to design controllers for high-dimensional nonlinear systems. The control is designed through the iterative linear quadratic regulator (ILQR), an algorithm that computes control by iteratively applying the linear quadratic regulator on the local linearization of the system at each time step. Since ILQR is computationally expensive, we propose to first construct reduced-order models (ROMs) of the high-dimensional nonlinear system. We derive nonlinear ROMs via projection, where the basis is computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT). Numerical experiments are performed on a semi-discretized nonlinear Burgers equation. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT, with BT-ROM based controllers outperforming LQG-BT slightly for very low-dimensional systems.

We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of the distributionally robust versions of these problems converge to the respective quantities of the original problems, as the sample size increases.

The problem of minimum-time, low-thrust, Earth-to-Mars interplanetary orbital trajectory optimization is considered. The minimum-time orbital transfer problem is modeled as a four-phase optimal control problem where the four phases correspond to planetary alignment, Earth escape, heliocentric transfer, and Mars capture. The four-phase optimal control problem is then solved using a direct collocation adaptive Gaussian quadrature collocation method. The following three models are used in the study: (1) circular planetary motion; (2) elliptic planetary motion; and (3) elliptic planetary motion with gravity perturbations, where the transfer begins in a geostationary orbit and terminates in a Mars-stationary orbit. Results for all three cases are provided, and one particular case is studied in detail to show the key features of the optimal solutions. Using the particular value thrust specific force of $0.00098\times 10^{-4}~\textrm{m}\cdot\textrm{s}^{-2}$, it was found that the minimum times for cases (1), (2), and (3) are, respectively, 215 d, 196 d, and 198 d with departure dates, respectively, of 1 July 2020, 30 June 2020, and 28 June 2020. Finally, the problem formulation developed in this study is compared against prior work on an Earth-to-Mars interplanetary orbit transfer where it is found that the results of this research show significant improvement in transfer time relative to the prior work., Comment: 41 pages, 8 figures, 7 tables

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces., 31 pages and 2 figures