The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important, but underappreciated, inequality.Comment: 20 page

Robust Secrecy Beamforming for MIMO SWIPT with Probabilistic Constraints

This paper considers simultaneous wireless information apower transfer (SWIPT) in a multiple-input multiple-output (MIMO) wiretapnd power transfer (SWIPT) in a multiple-input multiple-output (MIMO) wiretap channel with energy harvesting receivers. The main objective is to keep the probability of the legitimate user's achievable secrecy rate outage as well as the energy receivers' harvested energy outage as caused by CSI uncertainties below given thresholds. This probabilistic-constrained secrecy rate maximization problem presents a significant analytical and computational challenge since any closed-form for the probabilistic constraints with log-det functions is intractable. In this paper, we address this challenging issue using codeviation inequalitiesnvex restrictions. In particular, we derive decomposition-based large deviation inequalities to transform the probabilistic constraints into second-order cone (SOC) constraints which are easier to handle. Then we show that a robust safe solution can be obtained through solving two convex sub-problems in an alternating fashion

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Probabilistically Robust SWIPT for Secrecy MISOME Systems

This paper considers simultaneous wireless information and power transfer (SWIPT) in a multiple-input single-output (MISO) downlink system consisting of one multi-antenna transmitter, one single-antenna information receiver (IR), multiple multi-antenna eavesdroppers (Eves) and multiple single-antenna energy-harvesting receivers (ERs). The main objective is to keep the probability of the legitimate user's achievable secrecy rate outage as well as the ERs' harvested energy outage caused by channel state information (CSI) uncertainties below some prescribed thresholds. As is well known, the secrecy rate outage constraints present a significant analytical and computational challenge. Incorporating the energy harvesting (EH) outage constraints only intensifies that challenge. In this paper, we address this challenging issue using convex restriction approaches which are then proved to yield rank-one optimal beamforming solutions. Numerical results reveal the effectiveness of the proposed schemes.Comment: This is an open access article accepted for publication as a regular paper in the IEEE Transactions on Information Forensics & Security. Copyright (c) 2016 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]

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

Normas e estabilidade para modelos estocásticos cuja variação do controle e do estado aumentam a incerteza

Orientador: João Bosco Ribeiro do ValDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Essa dissertação de mestrado gira em torno da discussão sobre controle de sistemas incertos. Modelos matemáticos utilizados como base para o design de controladores automáticos são naturalmente uma representação aproximada do sistema real, o que, em conjunto com perturbações externas e dinâmica não modelada, gera incertezas a respeito dos sistemas estudados. Na literatura de controle, este tema vêm sendo discutido frequentemente, em particular nas sub-áreas de controle estocástico e controle robusto. Dentre as técnicas desenvolvidas dentro da teoria de controle estocástico, uma proposta recente se diferencia das demais por basear-se na idéia de que variações abruptas na política de controle possam acarretar em maiores incertezas a respeito do sistema. Matematicamente, essa noção é representada pelo uso de um ruído estocástico dependente do módulo da ação de controle, e a técnica foi apelidada de VCAI - acrônimo para variação do controle aumenta a incerteza. A definição da política de controle ótima correspondente, obtida por meio do método de programação dinâmica, mostra a existência de uma região ao redor do ponto de equilíbrio para a qual a política ótima é manter a ação de controle do equilíbrio inalterada, um resultado que parece particular à abordagem VCAI, mas que pode ser relacionado a políticas de gerenciamento cautelosas em áreas como economia e biologia. O problema de controle ótimo VCAI foi anteriormente resolvido ao adotar-se um critério de custo quadrático descontado e um horizonte de otimização infinito, e nessa dissertação nós utilizamos essa solução para atacar o problema de custo médio a longo prazo. Dada certa semelhança entre a estrutura do ruído estocástico na abordavem VCAI e modelos utilizados na teoria de controle robusto, discutimos ainda possíveis relações entre a abordagem proposta e controladores robustos. Discutimos ainda algumas possíveis aplicações do modelo propostoAbstract: This work discusses a new approach to the control of uncertain systems. Uncertain systems and their representation is a recurrent theme in control theory: approximate mathematical models, unmodeled dynamics and external disturbances are all sources of uncertainties in automated systems, and the topic has been extensively studied in the control literature, particularly within the stochastic and robust control research areas. Within the stochastic framework, a recent approach, named CVIU - control variation increases uncertainty, for short -, was recently proposed. The approach differs from previous models for assuming that a control action might actually increase the uncertainty about an unknown system, a notion represented by the use of stochastic noise depending on the absolute value of the control input. Moreover, the solution of the corresponding stochastic optimal control problem shows the existence of a region around the equilibrium point in which the optimal action is to keep the equilibrium control action unchanged. The CVIU control problem was previously solved by adopting a discounted quadratic cost formulation, and in this work we extend this previous result and study the corresponding long run average control problem. We also discuss possible relations between the CVIU approach and models from robust control theory, and present some potential applications of the theory presented hereMestradoAutomaçãoMestre em Engenharia Elétrica2016/02208-6, 2017/10340-4FAPES