International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Neural combinatorial optimization as an enabler technology to design real-time virtual network function placement decision systems

158 p.The Fifth Generation of the mobile network (5G) represents a breakthrough technology for thetelecommunications industry. 5G provides a unified infrastructure capable of integrating over thesame physical network heterogeneous services with different requirements. This is achieved thanksto the recent advances in network virtualization, specifically in Network Function Virtualization(NFV) and Software Defining Networks (SDN) technologies. This cloud-based architecture not onlybrings new possibilities to vertical sectors but also entails new challenges that have to be solvedaccordingly. In this sense, it enables to automate operations within the infrastructure, allowing toperform network optimization at operational time (e.g., spectrum optimization, service optimization,traffic optimization). Nevertheless, designing optimization algorithms for this purpose entails somedifficulties. Solving the underlying Combinatorial Optimization (CO) problems that these problemspresent is usually intractable due to their NP-Hard nature. In addition, solutions to these problems arerequired in close to real-time due to the tight time requirements on this dynamic environment. Forthis reason, handwritten heuristic algorithms have been widely used in the literature for achievingfast approximate solutions on this context.However, particularizing heuristics to address CO problems can be a daunting task that requiresexpertise. The ability to automate this resolution processes would be of utmost importance forachieving an intelligent network orchestration. In this sense, Artificial Intelligence (AI) is envisionedas the key technology for autonomously inferring intelligent solutions to these problems. Combining AI with network virtualization can truly transform this industry. Particularly, this Thesis aims at using Neural Combinatorial Optimization (NCO) for inferring endsolutions on CO problems. NCO has proven to be able to learn near optimal solutions on classicalcombinatorial problems (e.g., the Traveler Salesman Problem (TSP), Bin Packing Problem (BPP),Vehicle Routing Problem (VRP)). Specifically, NCO relies on Reinforcement Learning (RL) toestimate a Neural Network (NN) model that describes the relation between the space of instances ofthe problem and the solutions for each of them. In other words, this model for a new instance is ableto infer a solution generalizing from the problem space where it has been trained. To this end, duringthe learning process the model takes instances from the learning space, and uses the reward obtainedfrom evaluating the solution to improve its accuracy.The work here presented, contributes to the NCO theory in two main directions. First, this workargues that the performance obtained by sequence-to-sequence models used for NCO in the literatureis improved presenting combinatorial problems as Constrained Markov Decision Processes (CMDP).Such property can be exploited for building a Markovian model that constructs solutionsincrementally based on interactions with the problem. And second, this formulation enables toaddress general constrained combinatorial problems under this framework. In this context, the modelin addition to the reward signal, relies on penalty signals generated from constraint dissatisfactionthat direct the model toward a competitive policy even in highly constrained environments. Thisstrategy allows to extend the number of problems that can be addressed using this technology.The presented approach is validated in the scope of intelligent network management, specifically inthe Virtual Network Function (VNF) placement problem. This problem consists of efficientlymapping a set of network service requests on top of the physical network infrastructure. Particularly,we seek to obtain the optimal placement for a network service chain considering the state of thevirtual environment, so that a specific resource objective is accomplished, in this case theminimization of the overall power consumption. Conducted experiments prove the capability of theproposal for learning competitive solutions when compared to classical heuristic, metaheuristic, andConstraint Programming (CP) solvers

Advanced Optimization and Data-Driven Control in Smart Grid

The power grids are continuously evolving over the past decades, where new challenges and opportunities are embraced at the same time. On one hand, the penetration of renewable generations and other distributed energy resources (DER) is growing rapidly, whose different generation and control patterns could significantly impact the daily operation. On the other hand, the new communication, monitoring and regulating devices are gradually installed, which enable more control abilities of the generations, demands, and grids, and the feasibility to deploy more sophisticated control schemes.To leverage the new technique and overcome the new challenges in the smart girds, different optimization and control problems need to be solved for different roles including the system operator, demand, and financial traders. For the system operators, it is critical to maximizing the total social welfare while satisfying the operational constraints. To better coordinate the DER and improve the efficiency of distribution systems, the three-phase optimal power flow (OPF) problem algorithms are developed including the DCOPF algorithm for robustness and the ACOPF algorithm for optimality. Moreover, the deep reinforcement learning-based Volt-VAR control schemes are proposed to better maintain the voltage stability and electricity service quality.For demands resources, minimizing their energy bills will satisfy the energy needs is always their goal. Providing ancillary services by proactively adjusting their total demand is one of the potential choices. Through the provision of the services, the demands can not only receiving incentives from the system operators but also help to improve the reliability and stability of power grids. We develop control schemes specifically for the data centers to provide the phase balancing service in the distribution system and the frequency regulation service in the transmission system. The financial traders, it is desired to maximize their total profits. A better trading strategy with a more accurate forecast model can help increase the traders' gain and further improve the price convergence of the electricity market. Our machine learning based trading framework outperforms the existing approach and lays the foundation for market efficiency evaluation across the markets

Resource Management for Distributed Estimation via Sparsity-Promoting Regularization

Recent advances in wireless communications and electronics have enabled the development of low-cost, low-power, multifunctional sensor nodes that are small in size and communicate untethered in a sensor network. These sensor nodes can sense, measure, and gather information from the environment and, based on some local processing, they transmit the sensed data to a fusion center that is responsible for making the global inference. Sensor networks are often tasked to perform parameter estimation; example applications include battlefield surveillance, medical monitoring, and navigation. However, under limited resources, such as limited communication bandwidth and sensor battery power, it is important to design an energy-efficient estimation architecture. The goal of this thesis is to provide a fundamental understanding and characterization of the optimal tradeoffs between estimation accuracy and resource usage in sensor networks. In the thesis, two basic issues of resource management are studied, sensor selection/scheduling and sensor collaboration for distributed estimation, where the former refers to finding the best subset of sensors to activate for data acquisition in order to minimize the estimation error subject to a constraint on the number of activations, and the latter refers to seeking the optimal inter-sensor communication topology and energy allocation scheme for distributed estimation systems. Most research on resource management so far has been based on several key assumptions, a) independence of observation, b) strict resource constraints, and c) absence of inter-sensor communication, which lend analytical tractability to the problem but are often found lacking in practice. This thesis introduces novel techniques to relax these assumptions and provide new insights into addressing resource management problems. The thesis analyzes how noise correlation affects solutions of sensor selection problems, and proposes both a convex relaxation approach and a greedy algorithm to find these solutions. Compared to the existing sensor selection approaches that are limited to the case of uncorrelated noise or weakly correlated noise, the methodology proposed in this thesis is valid for any arbitrary noise correlation regime. Moreover, this thesis shows a correspondence between active sensors and the nonzero columns of an estimator gain matrix. Based on this association, a sparsity-promoting optimization framework is established, where the desire to reduce the number of selected sensors is characterized by a sparsity-promoting penalty term in the objective function. Instead of placing a hard constraint on sensor activations, the promotion of sparsity leads to trade-offs between estimation performance and the number of selected sensors. To account for the individual power constraint of each sensor, a novel sparsity-promoting penalty function is presented to avoid scenarios in which the same sensors are successively selected. For solving the proposed optimization problem, we employ the alternating direction method of multipliers (ADMM), which allows the optimization problem to be decomposed into subproblems that can be solved analytically to obtain exact solutions. The problem of sensor collaboration arises when inter-sensor communication is incorporated in sensor networks, where sensors are allowed to update their measurements by taking a linear combination of the measurements of those they interact with prior to transmission to a fusion center. In this thesis, a sparsity-aware optimization framework is presented for the joint design of optimal sensor collaboration and selection schemes, where the cost of sensor collaboration is associated with the number of nonzero entries of a collaboration matrix, and the cost of sensor selection is characterized by the number of nonzero rows of the collaboration matrix. It is shown that a) the presence of sensor collaboration smooths out the observation noise, thereby improving the quality of the signal and eventual estimation performance, and b) there exists a trade-off between sensor selection and sensor collaboration. This thesis further addresses the problem of sensor collaboration for the estimation of time-varying parameters in dynamic networks that involve, for example, time-varying observation gains and channel gains. Impact of parameter correlation and temporal dynamics of sensor networks on estimation performance is illustrated from both theoretical and practical points of view. Last but not least, optimal energy allocation and storage control polices are designed in sensor networks with energy-harvesting nodes. We show that the resulting optimization problem can be solved as a special nonconvex problem, where the only source of nonconvexity can be isolated to a constraint that contains the difference of convex functions. This specific problem structure enables the use of a convex-concave procedure to obtain a near-optimal solution

Neural combinatorial optimization as an enabler technology to design real-time virtual network function placement decision systems

158 p.The Fifth Generation of the mobile network (5G) represents a breakthrough technology for thetelecommunications industry. 5G provides a unified infrastructure capable of integrating over thesame physical network heterogeneous services with different requirements. This is achieved thanksto the recent advances in network virtualization, specifically in Network Function Virtualization(NFV) and Software Defining Networks (SDN) technologies. This cloud-based architecture not onlybrings new possibilities to vertical sectors but also entails new challenges that have to be solvedaccordingly. In this sense, it enables to automate operations within the infrastructure, allowing toperform network optimization at operational time (e.g., spectrum optimization, service optimization,traffic optimization). Nevertheless, designing optimization algorithms for this purpose entails somedifficulties. Solving the underlying Combinatorial Optimization (CO) problems that these problemspresent is usually intractable due to their NP-Hard nature. In addition, solutions to these problems arerequired in close to real-time due to the tight time requirements on this dynamic environment. Forthis reason, handwritten heuristic algorithms have been widely used in the literature for achievingfast approximate solutions on this context.However, particularizing heuristics to address CO problems can be a daunting task that requiresexpertise. The ability to automate this resolution processes would be of utmost importance forachieving an intelligent network orchestration. In this sense, Artificial Intelligence (AI) is envisionedas the key technology for autonomously inferring intelligent solutions to these problems. Combining AI with network virtualization can truly transform this industry. Particularly, this Thesis aims at using Neural Combinatorial Optimization (NCO) for inferring endsolutions on CO problems. NCO has proven to be able to learn near optimal solutions on classicalcombinatorial problems (e.g., the Traveler Salesman Problem (TSP), Bin Packing Problem (BPP),Vehicle Routing Problem (VRP)). Specifically, NCO relies on Reinforcement Learning (RL) toestimate a Neural Network (NN) model that describes the relation between the space of instances ofthe problem and the solutions for each of them. In other words, this model for a new instance is ableto infer a solution generalizing from the problem space where it has been trained. To this end, duringthe learning process the model takes instances from the learning space, and uses the reward obtainedfrom evaluating the solution to improve its accuracy.The work here presented, contributes to the NCO theory in two main directions. First, this workargues that the performance obtained by sequence-to-sequence models used for NCO in the literatureis improved presenting combinatorial problems as Constrained Markov Decision Processes (CMDP).Such property can be exploited for building a Markovian model that constructs solutionsincrementally based on interactions with the problem. And second, this formulation enables toaddress general constrained combinatorial problems under this framework. In this context, the modelin addition to the reward signal, relies on penalty signals generated from constraint dissatisfactionthat direct the model toward a competitive policy even in highly constrained environments. Thisstrategy allows to extend the number of problems that can be addressed using this technology.The presented approach is validated in the scope of intelligent network management, specifically inthe Virtual Network Function (VNF) placement problem. This problem consists of efficientlymapping a set of network service requests on top of the physical network infrastructure. Particularly,we seek to obtain the optimal placement for a network service chain considering the state of thevirtual environment, so that a specific resource objective is accomplished, in this case theminimization of the overall power consumption. Conducted experiments prove the capability of theproposal for learning competitive solutions when compared to classical heuristic, metaheuristic, andConstraint Programming (CP) solvers

Projection Robust Wasserstein Distance and Riemannian Optimization

Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work suggests that this quantity is more robust than the standard Wasserstein distance, in particular when comparing probability measures in high-dimensions. However, it is ruled out for practical application because the optimization model is essentially non-convex and non-smooth which makes the computation intractable. Our contribution in this paper is to revisit the original motivation behind WPP/PRW, but take the hard route of showing that, despite its non-convexity and lack of nonsmoothness, and even despite some hardness results proved by~\citet{Niles-2019-Estimation} in a minimax sense, the original formulation for PRW/WPP \textit{can} be efficiently computed in practice using Riemannian optimization, yielding in relevant cases better behavior than its convex relaxation. More specifically, we provide three simple algorithms with solid theoretical guarantee on their complexity bound (one in the appendix), and demonstrate their effectiveness and efficiency by conducing extensive experiments on synthetic and real data. This paper provides a first step into a computational theory of the PRW distance and provides the links between optimal transport and Riemannian optimization.Comment: Accepted by NeurIPS 2020; The first two authors contributed equally; fix the confusing parts in the proof and refine the algorithms and complexity bound

Bundle methods for regularized risk minimization with applications to robust learning

Supervised learning in general and regularized risk minimization in particular is about solving optimization problem which is jointly defined by a performance measure and a set of labeled training examples. The outcome of learning, a model, is then used mainly for predicting the labels for unlabeled examples in the testing environment. In real-world scenarios: a typical learning process often involves solving a sequence of similar problems with different parameters before a final model is identified. For learning to be successful, the final model must be produced timely, and the model should be robust to (mild) irregularities in the testing environment. The purpose of this thesis is to investigate ways to speed up the learning process and improve the robustness of the learned model. We first develop a batch convex optimization solver specialized to the regularized risk minimization based on standard bundle methods. The solver inherits two main properties of the standard bundle methods. Firstly, it is capable of solving both differentiable and non-differentiable problems, hence its implementation can be reused for different tasks with minimal modification. Secondly, the optimization is easily amenable to parallel and distributed computation settings; this makes the solver highly scalable in the number of training examples. However, unlike the standard bundle methods, the solver does not have extra parameters which need careful tuning. Furthermore, we prove that the solver has faster convergence rate. In addition to that, the solver is very efficient in computing approximate regularization path and model selection. We also present a convex risk formulation for incorporating invariances and prior knowledge into the learning problem. This formulation generalizes many existing approaches for robust learning in the setting of insufficient or noisy training examples and covariate shift. Lastly, we extend a non-convex risk formulation for binary classification to structured prediction. Empirical results show that the model obtained with this risk formulation is robust to outliers in the training examples

ON THE CONVERGENCE OF GRADIENT-LIKE FLOWS WITH NOISY GRADIENT INPUT

In view of solving convex optimization problems with noisy gradient input, we analyze the asymptotic behavior of gradient-like flows under stochastic disturbances. Specifically, we focus on the widely studied class of mirror descent schemes for convex programs with compact feasible regions, and we examine the dynamics' convergence and concentration properties in the presence of noise. In the vanishing noise limit, we show that the dynamics converge to the solution set of the underlying problem (a.s.). Otherwise, when the noise is persistent, we show that the dynamics are concentrated around interior solutions in the long run, and they converge to boundary solutions that are sufficiently "sharp". Finally, we show that a suitably rectified variant of the method converges irrespective of the magnitude of the noise (or the structure of the underlying convex program), and we derive an explicit estimate for its rate of convergence.Comment: 36 pages, 5 figures; revised proof structure, added numerical case study in Section