wait and judge scenario optimization we consider convex optimization problems with uncertain probabilistically described constraints in this context scenario optimization is a well recognized methodology where a sample of the constraints is used to describe uncertainty one says that the scenario solution generalizes well or has a high robustness level if it satisfies most of the other constraints besides those in the sample over the past 10 years the main theoretical investigations on the scenario approach have related the robustness level of the scenario solution to the number of optimization variables this paper breaks into the new paradigm that the robustness level is a posteriori evaluated after the solution is computed and the actual number of the so called support constraints is assessed wait and judge a new theory is presented which shows that a posteriori observing k support constraints in dimension d k allows one to draw conclusions close to those obtainable when the problem is from the outset in dimension k this new theory provides evaluations of the robustness that largely outperform those carried out based on the number of optimization variables

existence and uniqueness of pseudo almost periodic solutions of recurrent neural networks with time varying coefficients and mixed delays this paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to recurrent delayed neural networks several conditions guaranteeing the existence and uniqueness of such solutions are obtained in a suitable convex domain furthermore several methods are applied to establish sufficient criteria for the globally exponential stability of this system the approaches are based on constructing suitable lyapunov functionals and the well known banach contraction mapping principle moreover the attractivity and exponential stability of the pseudo almost periodic solution are also considered for the system a numerical example is given to illustrate the effectiveness of our results

an augmented lagrangian interior point method using directions of negative curvature we describe an efficient implementation of an interior point algorithm for non convex problems that uses directions of negative curvature these directions should ensure convergence to second order kkt points and improve the computational efficiency of the procedure some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified newton direction and the conditions to ensure feasibility of the iterates with respect to the simple bounds the use of multivariate barrier and penalty parameters is also discussed as well as the update rules for these parameters we analyze the convergence of the procedure both the linesearch and the update rule for the barrier parameter behave appropriately as the main goal of the paper is the practical usage of negative curvature a set of numerical results on small test problems is presented based on these results the relevance of using directions of negative curvature is discussed

notes on the scenario design approach the scenario optimization method developed in is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design that would otherwise be hard to tackle via standard deterministic techniques in this note we explore some further aspects of the scenario methodology and present two results pertaining to the tightness of the sample complexity bounds we also state a new theorem that enables the user to make a priori probabilistic claims on the scenario solution with one level of probability only

erratum to existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients by combining the theory of the exponential dichotomy and liapunov function we study the existence and attractivity of almost periodic solutions for cellular neural networks cnns with distributed delays and variable coefficientsdx idt b i t x i t j 1 na i j t f j x j t j 1 nb i j t f j m j 0 k i j u x j t u du i i t we obtain some sufficient conditions to ensure that for the networks there exists a unique almost periodic solution and all its solutions converge to such an almost periodic solution an example is given to illustrate that the conditions of our results are feasible

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

global stability analysis in delayed cellular neural networks in this paper the author analyzes further problems of global stability for a class of cellular neural networks with delays by means of the lyapunov functional method inequalities a 2 b 2 g 2ab and a 3 b 3 c 3 g 3abc a b cg 0 analysis technique some stability criteria are obtained under more general conditions these criteria can be applied to design globally stable networks and thus have important significance in both theory and applications

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

a feasible descent sqp algorithm for general constrained optimization without strict complementarity in this paper a class of optimization problems with equality and inequality constraints is discussed firstly the original problem is transformed to an associated simpler problem with only inequality constraints and a parameter the later problem is shown to be equivalent to the original problem if the parameter is large enough but finite then a feasible descent sqp algorithm for the simplified problem is presented at each iteration of the proposed algorithm a master direction is obtained by solving a quadratic program which always has a feasible solution with two corrections on the master direction by two simple explicit formulas the algorithm generates a feasible descent direction for the simplified problem and a height order correction direction which can avoid the maratos effect without the strict complementarity then performs a curve search to obtain the next iteration point thanks to the new height order correction technique under mild conditions without the strict complementarity the globally and superlinearly convergent properties are obtained finally an efficient implementation of the numerical experiments is reported

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

on the almost periodic solution of cellular neural networks with distributed delays by exponential dichotomy about differential equations a formal almost periodic solution aps of a class of cellular neural networks cnns with distributed delays is obtained then within different normed spaces several sufficient conditions guaranteeing the existence and uniqueness of an aps are proposed using two fixed point theorems based on the continuity property and some inequality techniques two theorems insuring the global stability of the unique aps are given comparing with known literatures all conclusions are drawn with slacker restrictions e g do not require the integral of the kernel function determining the distributed delays from zero to positive infinity to be one and the activation functions to be bounded etc besides all criteria are obtained by different ways finally two illustrative examples show the validity and that all criteria are easy to check and apply

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

sequential quadratic programming with penalization of the displacement in this paper we study the convergence of a sequential quadratic programming algorithm for the nonlinear programming problem the hessian of the quadratic program is the sum of an approximation of the lagrangian and of a multiple of the identity that allows us to penalize the displacement assuming only that the direction is a stationary point of the current quadratic program we study the local convergence properties without strict complementarity in particular we use a very weak condition on the approximation of the hessian to the lagrangian we obtain some global and superlinearly convergent algorithm under weak hypotheses as a particular case we formulate an extension of newton s method that is quadratically convergent to a point satisfying a strong sufficient second order condition

existence and global attractivity of almost periodic solutions for neural field with time delay in this paper the problem of existence and attractivity of almost periodic solutions for delayed neural field dnf with variable coefficients is discussed first the model of dnf is established as a modified neural field model secondly by combining the theory of the exponential dichotomy employing banach fixed point theory and using differential inequality technique some conditions for the the existence and attractivity of almost periodic solutions for the dnf are proposed

probabilistic robustness analysis explicit bounds for the minimum number of samples in this paper we study robustness analysis of control systems affected by bounded uncertainty motivated by the difficulty to perform this analysis when the uncertainty enters into the plant coefficients in a nonlinear fashion we study a probabilistic approach in this setting the uncertain parameters q are random variables bounded in a set q and described by a multivariate density function f q we then pose the following question given a performance level what is the probability that this level is attained the main content of this paper is to derive explicit bounds for the number of samples required to estimate this probability with a certain accuracy and confidence a priori specified it is shown that the number obtained is inversely proportional to these thresholds and it is much smaller than that of classical results finally we remark that the same approach can be followed to study several problems in a control system context for example we can evaluate the worst case h norm of the sensitivity function of multi input multi output systems or compute when the robustness margin is of interest

periodic oscillatory solution of bidirectional associative memory networks with delays periodic oscillatory phenomena of bidirectional associative memory bam networks with axonal signal transmission delays are investigated by constructing suitable lyapunov functionals and some analysis techniques some simple sufficient conditions are derived ensuring the existence and uniqueness of periodic oscillatory solutions of the bam with delays and all other solutions of the bam converge exponentially to a periodic oscillatory solution these conditions are presented in terms of system parameters and have an important leading significance in the design and applications of periodic oscillatory neural circuits for the bam with delays

existence and stability of almost periodic solution for bam neural networks with delays by using the banach fixed point theorem and constructing suitable lyapunov function some sufficient conditions are obtained ensuring existence uniqueness and global stability of almost periodic solution of the bam neural networks with variable coefficients and delays these results are helpful to design global exponential stable bam networks and almost periodic oscillatory bam networks

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

existence and exponential stability of periodic solution for impulsive cohen grossberg neural networks with time varying delays abstract this paper is concerned with the existence and global exponential stability of periodic solution for impulsive cohen grossberg neural networks with periodic coefficients and time varying delays some novel criteria on the existence and global exponential stability of periodic solution are derived by using impulsive differential inequality technique and the banach fixed point theorem the results obtained improve and extend a few of recent literature in some situations two examples with numerical simulations are given to demonstrate the efficiency of proposed methods

global dynamics of hopfield neural networks involving variable delays abstract in this article a model describing dynamics of hopfield neural networks involving variable delays is considered existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established further we derive sufficient criteria of global asymptotic stability gas of the equilibrium point

mixed time delays dependent exponential stability for uncertain stochastic high order neural networks this paper presents a discrete and distributed time delays dependent simultaneous approach to deterministic and uncertain stochastic high order neural networks new results are proposed in terms of linear matrix inequality lmi by exploiting a novel lyapunov krasovskii functional and by making use of novel techniques for time delay systems some constraints of systems are removed and new results cover some recently published works two numerical examples are given to show the usefulness of presented approach

nonconvex optimization using negative curvature within a modified linesearch this paper describes a new algorithm for the solution of nonconvex unconstrained optimization problems with the property of converging to points satisfying second order necessary optimality conditions the algorithm is based on a procedure which from two descent directions a newton type direction and a direction of negative curvature selects in each iteration the linesearch model best adapted to the properties of these directions the paper also presents results of numerical experiments that illustrate its practical efficiency

multiple almost periodic solutions in nonautonomous delayed neural networks a general methodology that involves geometric configuration of the network structure for studying multistability and multiperiodicity is developed we consider a general class of nonautonomous neural networks with delays and various activation functions a geometrical formulation that leads to a decomposition of the phase space into invariant regions is employed we further derive criteria under which the n neuron network admits 2n exponentially stable sets in addition we establish the existence of 2n exponentially stable almost periodic solutions for the system when the connection strengths time lags and external bias are almost periodic functions of time through applying the contraction mapping principle finally three numerical simulations are presented to illustrate our theory

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

tractable approximations to robust conic optimization problems in earlier proposals the robust counterpart of conic optimization problems exhibits a lateral increase in complexity i e robust linear programming problems lps become second order cone problems socps robust socps become semidefinite programming problems sdps and robust sdps become np hard we propose a relaxed robust counterpart for general conic optimization problems that a preserves the computational tractability of the nominal problem specifically the robust conic optimization problem retains its original structure i e robust lps remain lps robust socps remain socps and robust sdps remain sdps and b allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

existence and stability of almost periodic solutions in impulsive neural network models abstract the existence and global exponential stability of an almost periodic solution of an impulsive neural network model with distributed delays is considered in a matrix setting the approach transforms the original network into a matrix analysis problem where a set of sufficient conditions based on spectral radius is presented a concrete hopfield model shows the advantages in comparison with a classical norm approach

a note on the convergence of barrier algorithms to second order necessary points it has long been known that barrier algorithms for constrained optimization can produce a sequence of iterates converging to a critical point satisfying weak second order necessary optimality conditions when their inner iterations ensures that second order necessary conditions hold at each barrier minimizer we show that despite this strong second order necessary conditions may fail to be attained at the limit even if the barrier minimizers satisfy second order sufficient optimality conditions

existence and global attractivity of almost periodic solution for cellular neural network with distributed delays this paper is devoted to the existence and global attractivity of almost periodic solution for a class of cellular neural network with distributed delays and variable coefficients some sufficient conditions ensuring the existence and global attractivity of almost periodic solution are derived by employing banach fixed point theory and using differential inequality technique the results allow for the consideration of all unbounded neuron signal functions but not necessarily surjective thus these conditions obtained have highly important significance in designs and applications of the networks we extend and improve previously known results an example is also worked out to demonstrate the advantages of our results

exponential stability and periodic oscillatory solution in bam networks with delays both exponential stability and periodic oscillatory solution of bidirectional associative memory bam networks with axonal signal transmission delays are considered by constructing suitable lyapunov functional and some analysis techniques some simple sufficient conditions are given ensuring the global exponential stability and the existence of periodic oscillatory solutions of bam with delays these conditions are presented in terms of system parameters and have important leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for bam with delays in addition two examples are given to illustrate the results

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost

on the exponential stability and periodic solutions of delayed cellular neural networks a set of criteria is presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks dcnns by constructing suitable lyapunov functionals introducing many parameters and combining with the elementary inequality technique these criteria have important leading significance in the design and applications of globally stable dcnns and periodic oscillatory dcnns in addition earlier results are extended and improved other results are contained two examples are given to illustrate the theory