Corrigendum to “The envelope theorem for multiobjective convex programming via contingent derivatives” [J. Math. Anal. Appl. 372 (1) (2010) 197–207]

The aim of this note is to formulate an envelope theorem for vector convex programs. This version corrects an earlier work, “The envelope theorem for multiobjective convex programming via contingent derivatives” by Jiménez Guerra et al. (2010) [3]. We first propose a necessary and sufficient condition allowing to restate the main result proved in the alluded paper. Second, we introduce a new Lagrange multiplier in order to obtain an envelope theorem avoiding the aforementioned error.

Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Scalable sequential alternating proximal methods for sparse structural SVMs and CRFs

Structural Support Vector Machines (SSVMs) and Conditional Random Fields (CRFs) are popular discriminative methods used for classifying structured and complex objects like parse trees, image segments and part-of-speech tags. The datasets involved are very large dimensional, and the models designed using typical training algorithms for SSVMs and CRFs are non-sparse. This non-sparse nature of models results in slow inference. Thus, there is a need to devise new algorithms for sparse SSVM and CRF classifier design. Use of elastic net and L1-regularizer has already been explored for solving primal CRF and SSVM problems, respectively, to design sparse classifiers. In this work, we focus on dual elastic net regularized SSVM and CRF. By exploiting the weakly coupled structure of these convex programming problems, we propose a new sequential alternating proximal (SAP) algorithm to solve these dual problems. This algorithm works by sequentially visiting each training set example and solving a simple subproblem restricted to a small subset of variables associated with that example. Numerical experiments on various benchmark sequence labeling datasets demonstrate that the proposed algorithm scales well. Further, the classifiers designed are sparser than those designed by solving the respective primal problems and demonstrate comparable generalization performance. Thus, the proposed SAP algorithm is a useful alternative for sparse SSVM and CRF classifier design.

Extremum principle of crystal plasticity

An analytical method for determining slip shear rate under prescribed stress rate or prescribed strain rate has been presented on the basis of the incremental theory of crystal plasticity. The problem has been reduced to a quadric convex programming.In order to analyse the plastic response of crystals subjected to external load, two new extremum principles are proposed. They are equivalent to the boundary-value problem of crystal plasticity. By the new extremum principles, the slip shear rates are independent function which can be obtained from the variational equation.

Sorting under partial information (without the ellipsoid algorithm)

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical. We develop efficient algorithms for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithms differ in essential ways from theirs. Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed only once in a restricted class of graphs, permitting the use of a simpler algorithm. Specifically, we present: an O(n2) algorithm performing O(log n·log e(P)) comparisons; an O(n2.5) algorithm performing at most (1+ε) log e(P) + Oε(n) comparisons; an O(n2.5) algorithm performing O(log e(P)) comparisons. All our algorithms are simple to implement. © 2010 ACM.

Sequential anomaly detection in the presence of noise and limited feedback

This paper describes a methodology for detecting anomalies from sequentially observed and potentially noisy data. The proposed approach consists of two main elements: 1) filtering, or assigning a belief or likelihood to each successive measurement based upon our ability to predict it from previous noisy observations and 2) hedging, or flagging potential anomalies by comparing the current belief against a time-varying and data-adaptive threshold. The threshold is adjusted based on the available feedback from an end user. Our algorithms, which combine universal prediction with recent work on online convex programming, do not require computing posterior distributions given all current observations and involve simple primal-dual parameter updates. At the heart of the proposed approach lie exponential-family models which can be used in a wide variety of contexts and applications, and which yield methods that achieve sublinear per-round regret against both static and slowly varying product distributions with marginals drawn from the same exponential family. Moreover, the regret against static distributions coincides with the minimax value of the corresponding online strongly convex game. We also prove bounds on the number of mistakes made during the hedging step relative to the best offline choice of the threshold with access to all estimated beliefs and feedback signals. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality, as well as on the Enron email dataset. © 1963-2012 IEEE.

Design of Scheduling Algorithms Using Game Theoretic Ideas

Scheduling a set of jobs over a collection of machines to optimize a certain quality-of-service measure is one of the most important research topics in both computer science theory and practice. In this thesis, we design algorithms that optimize {\em flow-time} (or delay) of jobs for scheduling problems that arise in a wide range of applications. We consider the classical model of unrelated machine scheduling and resolve several long standing open problems; we introduce new models that capture the novel algorithmic challenges in scheduling jobs in data centers or large clusters; we study the effect of selfish behavior in distributed and decentralized environments; we design algorithms that strive to balance the energy consumption and performance.

The technically interesting aspect of our work is the surprising connections we establish between approximation and online algorithms, economics, game theory, and queuing theory. It is the interplay of ideas from these different areas that lies at the heart of most of the algorithms presented in this thesis.

The main contributions of the thesis can be placed in one of the following categories.

1. Classical Unrelated Machine Scheduling: We give the first polygorithmic approximation algorithms for minimizing the average flow-time and minimizing the maximum flow-time in the offline setting. In the online and non-clairvoyant setting, we design the first non-clairvoyant algorithm for minimizing the weighted flow-time in the resource augmentation model. Our work introduces iterated rounding technique for the offline flow-time optimization, and gives the first framework to analyze non-clairvoyant algorithms for unrelated machines.

2. Polytope Scheduling Problem: To capture the multidimensional nature of the scheduling problems that arise in practice, we introduce Polytope Scheduling Problem (\psp). The \psp problem generalizes almost all classical scheduling models, and also captures hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We design several competitive algorithms for the \psp problem and its variants for the objectives of minimizing the flow-time and completion time. Our work establishes many interesting connections between scheduling and market equilibrium concepts, fairness and non-clairvoyant scheduling, and queuing theoretic notion of stability and resource augmentation analysis.

3. Energy Efficient Scheduling: We give the first non-clairvoyant algorithm for minimizing the total flow-time + energy in the online and resource augmentation model for the most general setting of unrelated machines.

4. Selfish Scheduling: We study the effect of selfish behavior in scheduling and routing problems. We define a fairness index for scheduling policies called {\em bounded stretch}, and show that for the objective of minimizing the average (weighted) completion time, policies with small stretch lead to equilibrium outcomes with small price of anarchy. Our work gives the first linear/ convex programming duality based framework to bound the price of anarchy for general equilibrium concepts such as coarse correlated equilibrium.

Majorantes para a ordem de subgrafos induzidos k-regulares

Muitos dos problemas de otimização em grafos reduzem-se à determinação de um subconjunto de vértices de cardinalidade máxima que induza um subgrafo k-regular. Uma vez que a determinação da ordem de um subgrafo induzido k-regular de maior ordem é, em geral, um problema NP-difícil, são deduzidos novos majorantes, a determinar em tempo polinomial, que em muitos casos constituam boas aproximações das respetivas soluções ótimas. Introduzem-se majorantes espetrais usando uma abordagem baseada em técnicas de programação convexa e estabelecem-se condições necessárias e suficientes para que sejam atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro das matrizes de adjacência, laplaciana e laplaciana sem sinal. É ainda apresentado um algoritmo não polinomial para a determinação de umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular de ordem máxima para uma classe particular de grafos. Finalmente, faz-se um estudo computacional comparativo com vários majorantes e apresentam-se algumas conclusões.

New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range dependent, and distributeddelay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method

On relaxed LMI-based design for fuzzy controllers

Relaxed conditions for stability of nonlinear continuous-time systems given by fuzzy models axe presented. A theoretical analysis shows that the proposed method provides better or at least the same results of the methods presented in the literature. Digital simulations exemplify this fact. This result is also used for fuzzy regulators design. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno. The stability analysis and the design of controllers axe described by LMIs (Linear Matrix Inequalities), that can be solved efficiently using convex programming techniques.

On relaxed LMI-based designs for fuzzy regulators and fuzzy observers

Relaxed conditions for stability of nonlinear, continuous and discrete-time systems given by fuzzy models are presented. A theoretical analysis shows that the proposed methods provide better or at least the same results of the methods presented in the literature. Numerical results exemplify this fact. These results are also used for fuzzy regulators and observers designs. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno. The stability analysis and the design of controllers are described by linear matrix inequalities, that can be solved efficiently using convex programming techniques. The specification of the decay rate, constrains on control input and output are also discussed.

Robust State-Derivative Feedback LMI-Based Designs for Linear Descriptor Systems

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Aggregation-based decomposition for multi-divisional models

The aggregation theory of mathematical programming is used to study decentralization in convex programming models. A two-level organization is considered and a aggregation-disaggregation scheme is applied to such a divisionally organized enterprise. In contrast to the known aggregation techniques, where the decision variables/production planes are aggregated, it is proposed to aggregate resources allocated by the central planning department among the divisions. This approach results in a decomposition procedure, in which the central unit has no optimization problem to solve and should only average local information provided by the divisions.