Linear optimization

Background: In recent years, constrained optimization -usually referred to as flux balance analysis (FBA) -has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions. To augment the reliability of FBA-based flux calculations we propose an additional side constraint which assures thermodynamic realizability, i.e. that the flux directions are consistent with the corresponding changes of Gibb's free energies. The latter depend on metabolite levels for which plausible ranges can be inferred from experimental data. Computationally, our method results in the solution of a mixed integer linear optimization problem with quadratic scoring function. An optimal flux distribution together with a metabolite profile is determined which assures thermodynamic realizability with minimal deviations of metabolite levels from their expected values. We applied our novel approach to two exemplary metabolic networks of different complexity, the metabolic core network of erythrocytes (30 reactions) and the metabolic network iJR904 of Escherichia coli (931 reactions). Our calculations show that increasing network complexity entails increasing sensitivity of predicted flux distributions to variations of standard Gibb's free energy changes and metabolite concentration ranges. We demonstrate the usefulness of our method for assessing critical concentrations of external metabolites preventing attainment of a metabolic steady state. Our method incorporates the thermodynamic link between flux directions and metabolite concentrations into a practical computational algorithm. The weakness of conventional FBA to rely on intuitive assumptions about the reversibility of biochemical reactions is overcome. This enables the computation of reliable flux distributions even under extreme conditions of the network (e.g. enzyme inhibition, depletion of substrates or accumulation of end products) where metabolite concentrations may be drastically altered.

Background: Flux-balance analysis based on linear optimization is widely used to compute metabolic fluxes in large metabolic networks and gains increasingly importance in network curation and structural analysis. Thus, a computational tool flexible enough to realize a wide variety of FBA algorithms and able to handle batch series of flux-balance optimizations is of great benefit. We present FASIMU, a command line oriented software for the computation of flux distributions using a variety of the most common FBA algorithms, including the first available implementation of (i) weighted flux minimization, (ii) fitness maximization for partially inhibited enzymes, and (iii) of the concentration-based thermodynamic feasibility constraint. It allows batch computation with varying objectives and constraints suited for network pruning, leak analysis, flux-variability analysis, and systematic probing of metabolic objectives for network curation. Input and output supports SBML. FASIMU can work with free (lp_solve and GLPK) or commercial solvers (CPLEX, LINDO). A new plugin (faBiNA) for BiNA allows to conveniently visualize calculated flux distributions. The platform-independent program is an open-source project, freely available under GNU public license at including manual, tutorial, and plugins. Conclusions: We present a flux-balance optimization program whose main merits are the implementation of thermodynamics as a constraint, batch series of computations, free availability of sources, choice on various external solvers, and the flexibility on metabolic objectives and constraints.

Large-scale deployment of carbon capture and storage needs a dedicated infrastructure. Planning and designing of this infrastructure require incorporation of both temporal and spatial aspects. In this study, a toolbox has been developed that integrates ArcGIS, a geographical information system with spatial and routing functions, and MARKAL, an energy bottom-up model based on linear optimization. Application of this toolbox led to blueprints of a CO 2 infrastructure in the Netherlands. The results show that in a scenario with 20% and 50% CO 2 emissions reduction targets compared to their 1990 level in respectively 2020 and 2050, an infrastructure of around 600 km of CO 2 trunklines may need to be built before 2020. Investment costs for the pipeline construction and the storage site development amount to around 720 mV and 340 mV, respectively. The results also show the implication of policy choices such as allowing or prohibiting CO 2 storage onshore on CO 2 Capture and Storage (CCS) and infrastructure development. This paper illustrates how the ArcGIS/MARKAL-based toolbox can provide insights into a CCS infrastructure development, and support policy makers by giving concrete blueprints over time with respect to scale, pipeline trajectories, and deployment of individual storage sites.

A well studied and general setting for prediction and decision making is regret minimization in games. Originating independently in several disciplines, algorithms for regret minimization have proven to be empirically successful for a wide range of applications. Recently the design of algorithms for regret minimization in a wide array of settings has been influenced by tools from convex optimization. In this survey we describe two general methods for deriving algorithms and analyzing them, with a "convex optimization flavor". The methods we describe are general enough to capture most existing algorithms with single, simple and generic analysis, and lie at the heart of several recent advancements in prediction theory.

We introduce a new algorithm and a new analysis technique that is applicable to a variety of online optimization scenarios, including regret minimization for Lipschitz regret functions, universal portfolio management, online convex optimization and online utility maximization. In ...

Numerous machine learning problems require an exploration basis -a mechanism to explore the action space. We define a novel geometric notion of exploration basis with low variance called volumetric spanners, and give efficient algorithms to construct such bases. We show how efficient volumetric spanners give rise to an efficient and near-optimal regret algorithm for bandit linear optimization over general convex sets. Previously such results were known only for specific convex sets, or under special conditions such as the existence of an efficient self-concordant barrier for the underlying set.

Numerous machine learning problems require an exploration basis -a mechanism to explore the action space. We define a novel geometric notion of exploration basis with low variance called volumetric spanners, and give efficient algorithms to construct such bases. We show how efficient volumetric spanners give rise to an efficient and near-optimal regret algorithm for bandit linear optimization over general convex sets. Previously such results were known only for specific convex sets, or under special conditions such as the existence of an efficient self-concordant barrier for the underlying set. 1 although uniform sampling does at times work exceptionally well, e.g.

Building footprints have been shown to be extremely useful in urban planning, infrastructure development, and roof modeling. Current methods for creating these footprints are often highly manual and rely largely on architectural blueprints or skilled modelers. In this work we will use aerial LIDAR data to generate building footprints automatically. Existing automatic methods have been mostly unsuccessful due to large amounts of noise around building edges. We present a novel Bayesian technique for automatically constructing building footprints from a pre-classified LI-DAR point cloud. Our algorithm first computes a boundederror approximate building footprint using an application of the shortest path algorithm. We then determine the most probable building footprint by maximizing the posterior probability using linear optimization and simulated annealing techniques. We have applied our algorithm to more than 300 buildings in our data set and observe that we obtain accurate building footprints compared to the ground truth. Our algorithm is automatic and can be applied to other man-made shapes such as roads and telecommunication lines with minor modifications.

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound, but work very poorly in practice. To the contrary, the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this paper we discuss the new algorithmic variants and improved complexity results with respect to the new family of Self-Regular proximity based IPMs for Linear Optimization problems, and their generalizations to Conic and Semidefinite Optimization

This paper deals with the problem of dynamic modeling and identification of passenger cars. It presents a new method that is based on robotics techniques for modeling and description of tree-structured multibody systems. This method enables us to systematically obtain the dynamic identification model, which is linear with respect to the dynamic parameters. The estimation of the parameters is carried out using a weighted least squares method. The identification is tested using vehicle dynamics simulation software used by the car manufacturer PSA Peugeot-Citroën in order to define a set of trajectories with good excitation properties and to determine the number of degrees of freedom of the model. The method has then been used to estimate the dynamic parameters of an experimental Peugeot 406, which is equipped with different position, velocity, and force sensors.

The efficiency of urban transportation is getting more and more important because of the increasing rate of mobility demand. To plan, control and organize urban transportation in the most efficient way, we also need to consider the aspects of land use. To handle both of the mentioned urban planning areas together, we shall develop models, which are able to pay attention to all of their restrictive factors in the most simple way. It is a possible solution to simulate the urban area through a linear programming model.

The objective of this paper is to examine the driver’s mental workload experience during extreme nonlinear high slip driving conditions, i.e. with the tyres being close to saturation. This is done for the ISO Double Lane Change manoeuvre, based on extensive test data from two professional test drivers, for different speeds, and with varying tyres. Optimal double lane change closed loop performance is defined and determined and compared to the actual test performance, showing and explaining differences in this performance. A path-tracking driver model has been applied to examine the driver model parameters steering gain and preview time. It is shown that these parameters, which can easily be derived from the closed loop vehicle handling data, vary with mental workload as experienced by these test drivers. This correlation has been demonstrated in previous research for normal driving conditions, where workload, in this paper determined through RSME (Rating Scale Mental Effort) scores,...

Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the nonhomogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.

We study the problem of estimating vector-valued variables from noisy "relative" measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables. We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in R d , 1 ≤ d ≤ 3, that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable's estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithm-independent limits on the estimation accuracy achievable in large graphs.

This paper focus on offline energy management strategy based on dynamic losses computation made on accurate sources models. Hybrid energy systems (mainly Hybrid Electric Vehicle) should now be managed globally to reach the optimal operation minimizing a global cost criterion, such as hydrogen consumption. The presented operational research method is applied to a Fuel Cell based Vehicle (FCV) powertrain, and it has been adapted to the mathematical model depending on the sources characteristics (fuel cell and battery). In this paper losses and efficiency curves are issued from an accurate nonlinear model using experiments for both sources providing data representing the sources behavior to feed the optimization algorithm. Consumption comparisons on actual mission profiles demonstrate the optimal power splitting obtained.

Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. We are classifying all self-scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function. Together with a decomposition theorem for self-scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.

The goal of this study is to diagnose subclinical mastitis in cattle using multivariate classification applying images, Nuclear Magnetic Resonance (NMR) spectra and a bovine data survey set (DIB). The interest objects classification in milk images, NMR and DIB are processed with the algorithm of successive projections, linear discriminant analysis, and partial least squares discriminant analysis. The best diagnostic results obtained by image classification was 75% and 78% with NMR. As a result it can be observed that there is the possibility of classification for the diagnosis of mastitis by image and NMR, however, it is interesting to increase the number of samples for increasing the precision of two evaluated diagnosis methods.

Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a ε approximate solution is proportional to 1 ε 2 . Recently, Bienstock and Iyengar, following Nesterov, gave an algorithm for fractional packing linear programs which runs in 1 ε iterations. The latter algorithm requires to solve a convex quadratic program every iteration -an optimization subroutine which dominates the theoretical running time. We give an algorithm for convex programs with strictly convex constraints which runs in time proportional to 1 ε . The algorithm does not require to solve any quadratic program, but uses gradient steps and elementary operations only. Problems which have strictly convex constraints include maximum entropy frequency estimation, portfolio optimization with loss risk constraints, and various computational problems in signal processing. As a side product, we also obtain a simpler version of Bienstock and Iyengar's result for general linear programming, with similar running time. We derive these algorithms using a new framework for deriving convex optimization algorithms from online game playing algorithms, which may be of independent interest.

Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a ε approximate solution is proportional to 1 ε 2 . Recently, Bienstock and Iyengar, following Nesterov, gave an algorithm for fractional packing linear programs which runs in 1 ε iterations. The latter algorithm requires to solve a convex quadratic program every iteration -an optimization subroutine which dominates the theoretical running time. We give an algorithm for convex programs with strictly convex constraints which runs in time proportional to 1 ε . The algorithm does not require to solve any quadratic program, but uses gradient steps and elementary operations only. Problems which have strictly convex constraints include maximum entropy frequency estimation, portfolio optimization with loss risk constraints, and various computational problems in signal processing. As a side product, we also obtain a simpler version of Bienstock and Iyengar's result for general linear programming, with similar running time. We derive these algorithms using a new framework for deriving convex optimization algorithms from online game playing algorithms, which may be of independent interest.

Technical Report No. UCB/EECS-2008-18 http://www.eecs.berkeley.edu/Pubs/TechRpts/2008 /EECS-2008-18.html ... Copyright © 2008, by the author(s). All rights reserved. ... Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission.

The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal - Dual method is a Simplex - type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal - Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.

The determination of the factors influencing the effectiveness of algorithm visualization poses an interesting research question. In this paper, we present the results of a longitude empirical study regarding this question. The study was based on an evaluation of the Visual LinProg educational tool inside classrooms. Visual LinProg is a web-based educational tool, which solves linear programming problems using animation and visualization techniques. Visual LinProg was developed to be used in linear programming courses to supplement the teaching. Our empirical study is based on questionnaires that include quantitative and qualitative topics. This evaluation first indicates that Visual LinProg facilitates the learning of the revised simplex algorithm and second presents more results on factors influencing the understanding of this algorithm by the students/users of the Visual LinProg.

O conversor de uma unidade de craqueamento catalítico fluidizado (FCC) pode operar em dois possíveis modos: o modo de combustão completa e o modo de combustão parcial. Este trabalho tem como objetivo ordenar possíveis estruturas de controle para a operação do FCC nesses dois modos. A atenção maior nesse trabalho é dada à unidade de FCC empilhada (figura), tipo de unidade usada na Refinaria Alberto Pasqualini (Refap). A modelagem do riser foi feita utilizando um PFR. A cinética da reação de craqueamento se baseou em um modelo de 10 grupos que sub-divide os reatantes e os produtos em pseudo-componentes. O regenerador foi modelado com uma série de CSTR's, consideradas as reações de combustão total e parcial. O distúrbio mais significante é a composição da alimentação, o qual irá caracterizar a tendência de formar o coque. O modelo, escrito em linguagem C, foi implementado em Matlab com o uso do Simulink e seus resultados são comparados aos dados obtidos na planta industrial da Refap. Depois disso, serão achadas as melhores estruturas de controle. (RECOPE -FINEP).

For Cable-Driven Parallel Manipulators (CDPMs), employing redundant driving cables is necessary to obtain the full manipulation of the moving platform because of the unilateral driving property of the cables. Unlike rigid-link manipulators, the workspace of CDPMs is always determined and characterized by positive tension status of driving cables. In addition, it has been realized that the Tension Factor (TF) reflecting the relative tension distribution among the driving cables is an appropriate measure to evaluate the quality of tension restraint for CDPMs. However, since redundant cables are employed to drive the moving platform, the TF values are not unique for a particular moving platform pose. Therefore, how to determine the workspace and obtain the optimal TF value so as to generate a workspace with optimized performance become the major subjects of this paper. It is shown that the workspace can be generally formed from tension conditions verified by a recursive dimension-reduction approach and that the optimal TF value at every pose can be efficiently determined through a linear optimization approach, although it is essentially a nonlinear optimization problem. Computational examples are provided to demonstrate the effectiveness of the proposed algorithms.

The dynamic nature of a mobile ad hoc network (MANET) may result in a cluster of nodes being isolated from the rest of the network, especially when deployed in a terrain with blockages. To provide connectivity between the partitions of an ad hoc network that might occur due to mobility, a 'range extension' network can be employed. Such a network might consist of airborne communication platforms, or geostationary/low-earth-orbit satellites maintaining communication links with specific 'gateway' nodes that are dispersed among the mobile ground nodes. Thus, to communicate with a node that is geographically distant or belongs to a different network partition, an ad hoc node can relay its data packets through an appropriate mobile gateway and via the range extension network. In such an architecture, MANET is divided into different domains with a mobile gateway deployed for each domain. The objective, then, is to determine the position and trajectory of the gateways to optimize network performance metrics such as throughput and latency. In this paper, computation of the optimal position for a gateway is shown to be equivalent to a linear optimization problem by means of some simplifying but realistic assumptions. An algorithm is proposed for the control of the gateway trajectory. The practical constraints imposed by the velocity and maneuverability of the gateways are taken into account. Simulation results show a 10-15% improvement in the throughput and latency, per gateway domain, if a gateway has a dynamic trajectory whose locus follows the computed optimal position, as compared to a gateway that is statically placed at a fixed position, or to a gateway that has a random trajectory.

Strategic decision-making over valuable resources should consider risk-averse objectives. Many practical areas of application consider risk as central to decision- making. However, machine learning does not. As a result, research should provide insights and algorithms that endow machine learning with the ability to consider decision-theoretic risk. In particular, in estimating decision-theoretic risk on short dependent sequences generated from the most general possible class of processes for statistical inference and through decision-theoretic risk objectives in sequential decision-making. This thesis studies these two problems to provide principled algorithmic methods for considering decision-theoretic risk in machine learning. An algorithm with state-of-the-art performance is introduced for accurate estimation of risk statistics on the most general class of stationary–ergodic processes and risk-averse objectives are introduced in sequential decision-making (online learning) in bo...

In this paper, using the framework of self-regularity, we propose a hybrid adaptive algorithm for the linear optimization problem. If the current iterates are far from a central path, the algorithm employs a self-regular search direction, otherwise the classical Newton search direction is employed. This feature of the algorithm allows us to prove a worst case iteration bound. Our result matches the best iteration bound obtained by the pure self-regular approach and improves on the worst case iteration bound of the classical algorithm.

The Willshaw model is asymptotically the most efficient neural associative memory (NAM), but its finite version is hampered by high retrieval errors. Iterative retrieval has been proposed in a large number of different models to improve performance in auto-association tasks. In this paper, bidirectional retrieval for the hetero-associative memory task is considered: we define information efficiency as a general performance measure for bidirectional associative memory (BAM) and determine its asymptotic bound for the bidirectional Willshaw model. For the finite Willshaw model, an efficient new bidirectional retrieval strategy is proposed, the appropriate combinatorial model analysis is derived, and implications of the proposed sparse BAM for applications and brain theory are discussed. The distribution of the dendritic sum in the finite Willshaw model given by . Performance characteristics of associative nets. Network, 3, 407-414] allows no fast numerical evaluation. We derive a combinatorial formula with a highly reduced evaluation time that is used in the improved error analysis of the basic model and for estimation of the retrieval error in the naive model extension, where bidirectional retrieval is employed in the hetero-associative Willshaw model. The analysis rules out the naive BAM extension as a promising improvement. A new bidirectional retrieval algorithm -called crosswise bidirectional (CB) retrieval -is presented. The cross talk error is significantly reduced without employing more complex learning procedures or dummy augmentation in the pattern coding, as proposed in other refined BAM models [

This paper presents an approach to perform neutral facial image recognition through expression facial image using Hopfield neural networks. The dimension of neutral facial image is reduced and the image is sliced into gray levels, from the most significant bits to the less significant bits. The same process is applied to the expression facial image. The proposed approach using eight parallel Hopfield neural networks, one for each gray level, to recognition and recover the neutral facial image based in the expression facial image.

The efficiency of urban transportation is getting more and more important because of the increasing rate of mobility demand. To plan, control and organize urban transportation in the most efficient way, we also need to consider the aspects of land use. To handle both of the mentioned urban planning areas together, we shall develop models, which are able to pay attention to all of their restrictive factors in the most simple way. It is a possible solution to simulate the urban area through a linear programming model.

We consider a family of linear optimization problems over the ndimensional Klee-Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n -2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, while in practice typically only a few iterations, e.g., 50, suffices to obtain a high quality solution. Thus, the construction potentially exhibits the worst-case iterationcomplexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).

Prediction from expert advice is a fundamental problem in machine learning. A major pillar of the field is the existence of learning algorithms whose average loss approaches that of the best expert in hindsight (in other words, whose average regret approaches zero). Traditionally the regret of online algorithms was bounded in terms of the number of prediction rounds. Mansour and Stoltz [4] posed the question whether it is be possible to bound the regret of an online algorithm by the variation of the observed costs. In this paper we resolve this question, and prove such bounds in the fully adversarial setting, in two important online learning scenarios: prediction from expert advice, and online linear optimization.

Selecting the best hyperparameters for a particular optimization instance, such as the learning rate and momentum, is an important but nonconvex problem. As a result, iterative optimization methods such as hypergradient descent lack global optimality guarantees in general. We propose an online nonstochastic control methodology for mathematical optimization. First, we formalize the setting of meta-optimization, an online learning formulation of learning the best optimization algorithm from a class of methods. The meta-optimization problem over gradient-based methods can be framed as a feedback control problem over the choice of hyperparameters, including the learning rate, momentum, and the preconditioner. Although the original optimal control problem is nonconvex, we show how recent methods from online nonstochastic control using convex relaxations can be used to overcome the challenge of nonconvexity, and obtain regret guarantees against the best offline solution. This guarantees that in meta-optimization, given a sequence of optimization problems, we can learn a method that attains convergence comparable to that of the best optimization method in hindsight from a class of methods 1 .

We use the following mathematical notation in this writeup: • Vectors are denoted by boldface lower-case letters such as x ∈ R d . Coordinates of vectors are denoted by underscore notation x i or regular brackets x(i). • Matrices are denoted by boldface upper-case letters such as X ∈ R m×n . Their coordinates by X(i, j), or X ij . • Functions are denoted by lower case letters f : R d → R. • The k-th differential of function f is denoted by The gradient is denoted without the superscript, as ∇f . • We use the mathcal macro for sets, such as K ⊆ R d . • We denote the gradient at point x t as ∇ xt , or simply ∇ t . • We denote the global or local optima of functions by x . • We denote distance to optimality for iterative algorithms by h t = f (x t ) -f (x ). • Euclidean distance to optimality is denoted d t = x t -x .

We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online nonconvex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth [4], but theory for the analogous discrete time algorithms is left as an open problem. We prove an O(T 2 3 ) regret bound for non-convex online gradient descent in this setting, answering this open problem. Our analysis is based on a new and simple algorithmic equivalence method.

We consider the problem of controlling a possibly unknown linear dynamical system with adversarial perturbations, adversarially chosen convex loss functions, and partially observed states, known as non-stochastic control. We introduce a controller parametrization based on the denoised observations, and prove that applying online gradient descent to this parametrization yields a new controller which attains sublinear regret vs. a large class of closed-loop policies. In the fully-adversarial setting, our controller attains an optimal regret bound of $\sqrt{T}$-when the system is known, and, when combined with an initial stage of least-squares estimation, $T^{2/3}$ when the system is unknown; both yield the first sublinear regret for the partially observed setting. Our bounds are the first in the non-stochastic control setting that compete with \emph{all} stabilizing linear dynamical controllers, not just state feedback. Moreover, in the presence of semi-adversarial noise containing bo...

The online multi-armed bandit problem and its generalizations are repeated decision making problems, where the goal is to select one of several possible decisions in every round, and incur a cost associated with the decision, in such a way that the total cost incurred over all iterations is close to the cost of the best fixed decision in hindsight. The difference in these costs is known as the regret of the algorithm. The term bandit refers to the setting where one only obtains the cost of the decision used in a given iteration and no other information. Perhaps the most general form of this problem is the non-stochastic bandit linear optimization problem, where the set of decisions is a convex set in some Euclidean space, and the cost functions are linear. Only recently an efficient algorithm attaining Õ( √ T ) regret was discovered in this setting. In this paper we propose a new algorithm for the bandit linear optimization problem which obtains a regret bound of Õ( √ Q), where Q is the total variation in the cost functions. This regret bound, previously conjectured to hold in the full information case, shows that it is possible to incur much less regret in a slowly changing environment even in the bandit setting. Our algorithm is efficient and applies several new ideas to bandit optimization such as reservoir sampling.

Strategic decision-making over valuable resources should consider risk-averse objectives. Many practical areas of application consider risk as central to decision- making. However, machine learning does not. As a result, research should provide insights and algorithms that endow machine learning with the ability to consider decision-theoretic risk. In particular, in estimating decision-theoretic risk on short dependent sequences generated from the most general possible class of processes for statistical inference and through decision-theoretic risk objectives in sequential decision-making. This thesis studies these two problems to provide principled algorithmic methods for considering decision-theoretic risk in machine learning. An algorithm with state-of-the-art performance is introduced for accurate estimation of risk statistics on the most general class of stationary–ergodic processes and risk-averse objectives are introduced in sequential decision-making (online learning) in bo...

We consider the problem of online optimization, where a learner chooses a decision from a given decision set and suffers some loss associated with the decision and the state of the environment. The learner's objective is to minimize its cumulative regret against the best fixed decision in hindsight. Over the past few decades numerous variants have been considered, with many algorithms designed to achieve sub-linear regret in the worst case. However, this level of robustness comes at a cost. Proposed algorithms are often over-conservative, failing to adapt to the actual complexity of the loss sequence which is often far from the worst case. In this paper we introduce a general algorithm that, provided with a "safe" learning algorithm and an opportunistic "benchmark", can effectively combine good worst-case guarantees with much improved performance on "easy" data. We derive general theoretical bounds on the regret of the proposed algorithm and discuss its implementation in a wide range of applications, notably in the problem of learning with shifting experts (a recent COLT open problem). Finally, we provide numerical simulations in the setting of prediction with expert advice with comparisons to the state of the art.

In this paper we measure how much a linear optimization problem, in R n , has to be perturbed in order to lose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed those problems in the boundary of the set of solvable ones, then we can say that this paper deals with the associated distance to ill-posedness. Our parameter space is the set of all the linear semi-infinite programming problems with a fixed, but arbitrary, index set. In this framework, which includes as a particular case the ordinary linear programming, we obtain a formula for the distance from a solvable problem to unsolvability in terms of the nominal problem's coefficients. Moreover, this formula also provides the exact expression, or a lower bound, of the distance from an unsolvable problem to solvability. The relationship between the solvability and the primal-dual consistency is analyzed in the semi-infinite context, underlining the differences with the finite case.

paper introduces a procedure for the design of modified linear-quadratic (LQ) state-feedback controls that tolerate actuator outages. The controls improve on the known stability gain-margin properties of the standard LQ regulator by tolerating the insertion of any independent gains from zero to infinity into selected feedback loops. They also guarantee a given performance bound despite the insertion of gains from zero to two into those loops. The reliable LQ design is shown to be equivalent to a standard LQ-optimal design with a modified performance index. Thus, the design procedure is seen as a means of choosing a particular quadratic performance index for which the optimal control will possess the desired reliability properties.

paper introduces a procedure for the design of modified linear-quadratic (LQ) state-feedback controls that tolerate actuator outages. The controls improve on the known stability gain-margin properties of the standard LQ regulator by tolerating the insertion of any independent gains from zero to infinity into selected feedback loops. They also guarantee a given performance bound despite the insertion of gains from zero to two into those loops. The reliable LQ design is shown to be equivalent to a standard LQ-optimal design with a modified performance index. Thus, the design procedure is seen as a means of choosing a particular quadratic performance index for which the optimal control will possess the desired reliability properties.

Maintenance activities can be performed throughout the lifetime of a particular facility or piece of equipment, thereby affecting its quality in a continuous fashion. It is assumed in this paper that quality of a facility is determined by natural factors, rate of use and maintenance investments while demand for the facility is assumed to be a function of its quality. A dynamic optimization model in the form of a simple linear optimal control problem is then developed in order to determine optimal maintenance policies under these circumstances. Bang-bang and singular policies are derived and given economic interpretations. An algorithm is presented for numerical solution of the Pontryagin necessary conditions. Finally, sufficiency conditions are obtained for the model structure considered.

This paper presents a method for the automatic detection of malfunctioning Traffic Count Stations (TCS) in a transport system. First, double linear optimization is used to detect inadmissible errors in the recordings of a series of TCS and next, the TCS that are most likely to be failing are identified. The method has been applied to an urban traffic network showing success rates up to 93% in identifying the TCS that are failing.

This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central path interior point algorithm for the the entropic universal barrier function. This connection exhibits several benefits. First, we are able improve the state of the art time complexity for convex optimization under the membership oracle model. We improve the analysis of the randomized algorithm of Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii [16] that underly the central path following interior point algorithm. We are able to tighten the temperature schedule for simulated annealing which gives an improved running time, reducing by square root of the dimension in certain instances. Second, we get an efficient randomized interior point method with an efficiently computable universal barrier for any convex set described by a membership oracle. Previously, efficiently computable barriers were known only for particular convex sets.

Prediction from expert advice is a fundamental problem in machine learning. A major pillar of the field is the existence of learning algorithms whose average loss approaches that of the best expert in hindsight (in other words, whose average regret approaches zero). Traditionally the regret of online algorithms was bounded in terms of the number of prediction rounds. Cesa-Bianchi, Mansour and Stoltz (Mach. Learn. 66(2-3):21-352, 2007) posed the question whether it is be possible to bound the regret of an online algorithm by the variation of the observed costs. In this paper we resolve this question, and prove such bounds in the fully adversarial setting, in two important online learning scenarios: prediction from expert advice, and online linear optimization.