robust two stage stochastic linear optimization with risk aversion abstract we study a two stage stochastic linear optimization problem where the recourse function is risk averse rather than risk neutral in particular we consider the mean conditional value at risk objective function in the second stage the model is robust in the sense that the distribution of the underlying random variable is assumed to belong to a certain family of distributions rather than to be exactly known we start from analyzing a simple case where uncertainty arises only in the objective function and then explore the general case where uncertainty also arises in the constraints we show that the former problem is equivalent to a semidefinite program and the latter problem is generally np hard applications to two stage portfolio optimization material order problems stochastic production transportation problem and single facility minimax distance problem are considered numerical results show that the proposed robust risk averse two stage stochastic programming model can effectively control the risk with solutions of acceptable good quality

an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

a connection between normalized coprime factorizations and linear quadratic regulator theory given a transfer matrix described by a minimal state space triple a method is given for computing state space realizations for the numerator and denominator of a normalized stable right coprime factorization for the transfer matrix the method involves the solution of an algebraic riccati equation it allows the use of existing computational state space algorithms in finding normalized stable right coprime factorizations and avoids explicit calculations of spectral factors

distributionally robust stochastic dual dynamic programming we consider a multistage stochastic linear program that lends itself to solution by stochastic dual dynamic programming sddp in this context we consider a distributionally robust variant of the

scheduling arrivals to a stochastic service delivery system using copositive cones in this paper we investigate a stochastic appointment scheduling problem in an outpatient clinic with a single doctor the number of patients and their sequence of arrivals are fixed and the scheduling problem is to determine an appointment time for each patient the service durations of the patients are stochastic and only the mean and covariance estimates are known we do not assume any exact distributional form of the service durations and we solve for distributionally robust schedules that minimize the expectation of the weighted sum of patients waiting time and the doctor s overtime we formulate this scheduling problem as a convex conic optimization problem with a tractable semidefinite relaxation our model can be extended to handle additional support constraints of the service durations using the primal dual optimality conditions we prove several interesting structural properties of the optimal schedules we develop an efficient semidefinite relaxation of the conic program and show that we c

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

the robust regulation problem with robust stability a subspace valued function approach among the most common purposes are the tracking of reference signals and the rejection of disturbance signals in the face of uncertainties the related design problem is called the robust regulation problem here we investigate the tradeoff between the robust regulation constraint and the requirement of robust stability with respect to perturbations in the normalized coprime factors of the plant we use an approach in terms of subspace valued functions associated to plant and controller in this way necessary and sufficient conditions for robust regulation with robust stability can be related to a simple geometric property

models for minimax stochastic linear optimization problems with risk aversion we propose a semidefinite optimization sdp model for the class of minimax two stage stochastic linear optimization problems with risk aversion the distribution of second stage random variables belongs to a set of multivariate distributions with known first and second moments for the minimax stochastic problem with random objective we provide a tight sdp formulation the problem with random right hand side is np hard in general in a special case the problem can be solved in polynomial time explicit constructions of the worst case distributions are provided applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach in our experiments the performance of minimax solutions is close to that of data driven solutions under the multivariate normal distribution and better under extremal distributions the minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity

distributionally robust multi item newsvendor problems with multimodal demand distributions we present a risk averse multi dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed the demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description we assume that the newsvendor hedges against distributional ambiguity by minimizing the worst case risk of the order portfolio over all distributions that are compatible with the given modality information we demonstrate that the resulting distributionally robust optimization problem is mathrm np np hard but admits an efficient numerical solution in quadratic decision rules this approximation is conservative and computationally tractable moreover it achieves a high level of accuracy in numerical tests we further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments

distribution free inventory risk pooling in a multi location newsvendor we study a multilocation newsvendor network when the only information available on the joint distribution of demands are the values of its mean vector and covariance matrix we adopt a distribution

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

convexity and decomposition of mean risk stochastic programs traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion a common approach to addressing risk in decision making problems is to consider a weighted mean risk objective where some dispersion statistic is used as a measure of risk we investigate the computational suitability of various mean risk objective functions in addressing risk in stochastic programming models we prove that the classical mean variance criterion leads to computational intractability even in the simplest stochastic programs on the other hand a number of alternative mean risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms we propose decomposition based parametric cutting plane algorithms to generate mean risk efficient frontiers for two particular classes of mean risk objectives

the value of stochastic modeling in two stage stochastic programs with cost uncertainty although stochastic programming is probably the most effective framework for handling decision problems that involve uncertain variables it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables in practice this might require one to collect a large amount of observations to consult with experts of the specialized field of practice or to make simplifying assumptions about the underlying system when none of these options seem feasible a common heuristic has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value otherwise known as the solution of the mean value problem in this paper we show that when 1 the stochastic program takes the form of a two stage mixed integer stochastic linear programs and 2 the uncertainty is limited to the objective function the solution of the mean value problem is in fact robust with respect to the selection of a stochastic model we also propose tractable methods that will bound the actual value of stochastic modeling i e how much improvement can be achieved by investing more efforts in the resolution of the stochastic model our framework is applied to an airline fleet composition problem in the three cases that are considered our results indicate that resolving the stochastic model can not lead to more than a 7 improvement of expected profits thus providing arguments against the need to develop these more sophisticated models

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

remarks on order reduction for a robustly suboptimal controller via singular perturbations abstract it is shown that under a generic assumption the suboptimally robust controller constructed according to glover and mcfarlane 1989 leads to a singularly perturbed system which may be reduced in the usual way avoiding thus the difficulties mentioned by habets 1991 the reduced controller obtained by this procedure is in fact optimal

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

identification of normalized coprime factors through constrained curve fitting this paper deals with identifying finite dimensional normalized right coprime factors nrcfs under a stochastic framework from a sequence of their frequency response estimates a two step procedure is suggested for the estimation in which two cost functions are sequentially minimized these cost functions are constructed on the basis of weighted curve fitting and the conditions on nrcfs a numerical procedure is proposed for computing the estimate moreover it is proved that under some conditions the estimate converges with probability 1 w p 1 to its actual value with increasing the data length and in general the optimal estimate is asymptotically approximately gaussian the efficiency of the proposed method in nrcf identification is illustrated by numerical simulations

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

two stage stochastic linear programs with incomplete information on uncertainty abstract two stage stochastic linear programming is a classical model in operations research the usual approach to this model requires detailed information on distribution of the random variables involved in this paper we only assume the availability of the first and second moments information of the random variables by using duality of semi infinite programming and adopting a linear decision rule we show that a deterministic equivalence of the two stage problem can be reformulated as a second order cone optimization problem preliminary numerical experiments are presented to demonstrate the computational advantage of this approach

ambiguous risk constraints with moment and unimodality information optimization problems face random constraint violations when uncertainty arises in constraint parameters effective ways of controlling such violations include risk constraints e g chance constraints and conditional value at risk constraints this paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous in particular we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality we find that the ambiguous risk constraints in this setting can be recast as a set of second order cone soc constraints in order to facilitate the algorithmic implementation we also derive efficient ways of finding violated soc constraints finally we demonstrate the theoretical results via computational case studies on power system operations

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

some results on graph spectra abstract this paper presents a variety of results on graph spectra the number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix we establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors it is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement we generalize to any eigenvalue a condition for 0 and 1 to be eigenvalues of a graph and its complement respectively finally we generalize to non simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

mixed 0 1 linear programs under objective uncertainty a completely positive representation in this paper we analyze mixed 0 1 linear programs under objective uncertainty the mean vector and the second moment matrix of the nonnegative objective coefficients are assumed to be known but the exact form of the distribution is unknown our main result shows that computing a tight upper bound on the expected value of a mixed 0 1 linear program in maximization form with random objective is a completely positive program this naturally leads to semidefinite programming relaxations that are solvable in polynomial time but provide weaker bounds the result can be extended to deal with uncertainty in the moments and more complicated objective functions examples from order statistics and project networks highlight the applications of the model our belief is that the model will open an interesting direction for future research in discrete and linear optimization under uncertainty

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

optimal h multivariable robust controllers and the relationship to lqg design problems a new solution for the multivariable minimax optimal robust design problem is presented the relationship between lqg and h design methods is considered and the h solution is obtained from the solution of a related lqg problem the approach provides insights into the relationship between the h and lqg design philosophies the h solution involves the same type of steps which are involved in the polynomial based solutions of the lqg problem namely spectral factorization and the solution of linear diophantine equations a simplified design procedure is described which requires only the solution of a standard eigenvalue eigenvector problem

applying the minimax criterion in stochastic recourse programs we consider an optimization problem in which some uncertain parameters are replaced by random variables the minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions in this paper we establish a number of stability results for the minimax stochastic program justifying in particular the approach of restricting attention to probability measures with support in some known finite set following this approach we elaborate solution procedures for the minimax problem in the setting of two stage stochastic recourse models considering the linear recourse case as well as the integer recourse case since the solution procedures are modifications of well known algorithms their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments

second order lower bounds on the expectation of a convex function we develop a class of lower bounds on the expectation of a convex function the bounds utilize the first two moments of the underlying random variable whose support is contained in a bounded interval or hyperrectangle our bounds have applications to stochastic programs whose random parameters are known only through limited moment information computational results are presented for two stage stochastic linear programs

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh

robustness to dependency in portfolio optimization using overlapping marginals in this paper we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses for a fr echet class of discrete distributions with overlapping marginals we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming to guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information we make use of a graph theoretic property known as the running intersection property building on this property we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst case conditional value at risk measure lastly we use a data driven approach with financial return data to identify the fr echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample based approach

distributionally robust optimization with matrix moment constraints lagrange duality and cutting plane methods a key step in solving minimax distributionally robust optimization dro problems is to reformulate the inner maximization w r t probability measure as a semiinfinite programming problem through lagrange dual slater type conditions have been widely used for strong duality zero dual gap when the ambiguity set is defined through moments in this paper we investigate effective ways for verifying the slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem moreover we propose two discretization schemes for solving the dro with one for the dualized dro and the other directly through the ambiguity set of the dro in the absence of strong duality the discretization scheme via lagrange duality may provide an upper bound for the optimal value of the dro whereas the direct discretization approach provides a lower bound two cutting plane schemes are consequently proposed one for the discretized dualized dro and the other for the minimax dro with discretized ambiguity set convergence analysis is presented for the approximation schemes in terms of the optimal value optimal solutions and stationary points comparative numerical results are reported for the resulting algorithms

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

a note on hoffman type identities of graphs an eigenvalue of a graph g is called main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero hoffman a j hoffman on the polynomial of a graph amer math monthly 70 1963 30 36 proved that g is a connected k regular graph if and only if n i 2t a ii i 2t k i j where i is the unit matrix and j the all one matrix and 1 k 2 t are all distinct eigenvalues of adjacency matrix a g in this note some generalizations of hoffman identity are presented by means of main eigenvalues

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

convex optimal uncertainty quantification optimal uncertainty quantification ouq is a framework for numerical extreme case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution this paper presents sufficient conditions under which an ouq problem can be reformulated as a finite dimensional convex optimization problem for which efficient numerical solutions can be obtained the sufficient conditions include that the objective function is piecewise concave and the constraints are piecewise convex in particular we show that piecewise concave objective functions may appear in applications where the objective is defined by the optimal value of a parameterized linear program

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

integer programming approaches for appointment scheduling with random no shows and service durations we consider a single server scheduling problem given a fixed sequence of appointment arrivals with random no shows and service durations the probability distribution of the uncertain parameters is assumed to be ambiguous and only the support and first moments are known we formulate a class of distributionally robust dr optimization models that incorporate the worst case expectation conditional value at risk penalty cost of appointment waiting server idleness and overtime into the objective or constraints our models flexibly adapt to different prior beliefs of no show uncertainty we obtain exact mixed integer nonlinear programming reformulations and derive valid inequalities to strengthen the reformulations that are solved by decomposition algorithms in particular we derive convex hulls for special cases of no show beliefs yielding polynomial sized linear programming models for the least and the most conservative supports of no shows we test various instances to demonstrate the computational e

exact computation of infimum for a class of continuous time h optimal control problem with a nonzero direct feedthrough term from the disturbance input to the controlled output a noniterative method for the computation of infimum for a class of continuous time h optimal control problem is considered in this paper the problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems the method is applicable to systems where i the transfer function from the disturbance input to the measurement output is free of imaginary axis invariant zeros and left invertible and ii the transfer function from the control input to the controlled output of the given system is free of imaginary axis invariant zeros and right invertible the result presented in this paper is a continuation of the previous work of the author and his co workers chen et al 1992 in which the direct feedthrough term from the disturbance input to the measurement output of the given system are required to be zero

process flexibility a distribution free bound on the performance of k chain process flexibility has been widely applied in many industries as a competitive strategy to improve responsiveness to demand uncertainty an important flexibility concept is the long chain proposed by jordan and graves 1995 jordan wc graves sc 1995 principles on the benefits of manufacturing process flexibility management sci 41 4 577 594 the effectiveness of the long chain has been investigated via numerical as well as theoretical analysis for specific probability distributions of the random demand in this paper we obtain in closed form a distribution free bound on the ratio of the expected sales of the long chain relative to that of full flexibility our bound depends only on the mean and standard deviation of the random demand but compares very well with the ratio that uses complete information of the demand distribution this suggests the robustness of the performance of the long chain under different distributions we also prove a similar result for k chain a more general flexibility

smooth weight optimization in ℋ loop shaping design abstract smooth variation in the magnitude response of weights facilitates loop shaping design as it prevents the cancellation of important modes of the system for example lightly damped poles zeros of flexible structures when the shaped plant is formed based on closed loop design specifications for accurate fitting of transfer functions to magnitude data smooth weights also allow low order transfer functions to be used when such smooth variations in their magnitude responses are computed point wise in frequency in this paper smoothness constraints for weights expressed as gradient constraints on a log scale in db decade which is intuitive from a design perspective are imposed in a weight optimization framework for loop shaping control this work builds on a lanzon weight optimization in loop shaping automatica 41 1 2005 1201 1208 where additional constraints are formulated in linear matrix inequality lmi form to cast a complete weight optimization framework the resulting algorithm thus maximizes the robust stability margin and simultaneously synthesizes smooth weights along with a stabilizing controller a numerical example is given to elucidate the efficacy of the smoothness constraints in loop shaping control

coprime factorization and optimal control on the doubly infinite discrete time axis we study the problem of strongly coprime factorization over h infinity of the unit disc we give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete time axis in particular we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic riccati equation of an arbitrary realization have a solution in general unbounded and even nondensely defined and that a coupling condition involving these solutions is satisfied

models and algorithms for distributionally robust least squares problems we present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems these probability ambiguity descriptions are given by 1 confidence region over the first two moments 2 bounds on the probability measure with moments constraints 3 the kantorovich probability distance from a given measure for the first case we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms for the second case we derive the equivalent lagrangian problem and show that it is a convex stochastic programming problem we further analyze three special subcases i finite support ii measure bounds by a reference probability measure iii measure bounds by two reference probability measures with known density functions we show that case i has an equivalent semidefinite programming reformulation and the sample average approximations of case ii and iii have equivalent semidefinite programming reformulations for ambiguity description 3 we show that the finite support case can be solved by using an equivalent second order cone programming reformulation

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

choice prediction with semidefinite optimization when utilities are correlated we consider the problem of making choice prediction by optimizing the expectation of maximum utility in discrete choice situations and propose a discrete choice model which generates choice probabilities through a semidefinite program the choice model termed as the cross moment model cmm is parsimonious in that it uses only the mean and covariance information of the utilities it is encouraging that cmm generates reasonable choice estimates when utilities are correlated even though no distributional assumptions on random utilities are made we present a few examples in route choice setting and random walk to test the quality of choice prediction using cmm by capturing correlations among utilities cmm avoids some of the common behavioral limitations such as the independence of irrelevant alternatives and the invariate proportion of substitution present in several discrete choice models being a convex optimization problem it obviates the use of exhaustive simulation in computing choice probabilities for which the multinomial probit mnp model is often criticized cmm can be easily used in design problems such as product line selection and assortment planning we exploit cmm to solve a flexible packaging design problem faced by online retailers and warehouses we use the data provided by a local service part supplier for this design problem and compare the results with multinomial logit and mnp models we find that cmm not only captures utility correlations in this problem and provides good designs but also has computational advantages over mnp which uses simulation

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

gap metric problem for mimo delay systems parametrization of all suboptimal controllers we consider the problem of robustness optimization in the gap metric for mimo systems with a scalar time delay we present an algorithm for the parametrization of all suboptimal controllers in our algorithm the aak theory plays the central role using this approach suboptimal controllers can be found by computing solutions to certain infinite rank operator equations we show that these equations can be solved numerically by modifying the two point boundary value problem approach reported earlier for the computation of optimal robustness radius we present a numerical example to illustrate the procedure

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

quadratic two stage stochastic optimization with coherent measures of risk a new scheme to cope with two stage stochastic optimization problems uses a risk measure as the objective function of the recourse action where the risk measure is defined as the worst case expected values over a set of constrained distributions this paper develops an approach to deal with the case where both the first and second stage objective functions are convex linear quadratic it is shown that under a standard set of regularity assumptions this two stage quadratic stochastic optimization problem with measures of risk is equivalent to a conic optimization problem that can be solved in polynomial time

hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs abstract for a nonnegative n n matrix a we find that there is a polynomial f x r x such that f a is a positive matrix of rank one if and only if a is irreducible furthermore we show that the lowest degree such polynomial f x with tr f a n is unique thus generalizing the well known definition of the hoffman polynomial of a strongly connected regular digraph for any irreducible nonnegative n n matrix a we are led to define its hoffman polynomial to be the polynomial f x of minimum degree satisfying that f a is positive and has rank 1 and trace n the hoffman polynomial of a strongly connected digraph is defined to be the hoffman polynomial of its adjacency matrix we collect in this paper some basic results and open problems related to the concept of hoffman polynomials

quantitative stability of two stage distributionally robust risk optimization problem with full random linear semi definite recourse abstract in this paper we study a distributionally robust risk optimization drro problem where the information on the probability distribution of the underlying random variables is incomplete but it is possible to use partial information to construct an ambiguity set of probability distributions in some cases decision vector x may have a direct impact on the likelihood of the underlying random events that occur after the decision is taken which motivates us to propose an ambiguity set to be parametric and decision dependent to conduct quantitative stability analysis of the optimal value function and the optimal solution mapping of the drro problem we derive error bounds results for the parametrized ambiguity set under the total variation metric and investigate lipschitz continuity of the objective function of the drro problem under some conditions as an application we demonstrate that the two stage stochastic linear semi definite programs satisfy these conditions and then apply results obtained to it

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

robust sample average approximation sample average approximation saa is a widely popular approach to data driven decision making under uncertainty under mild assumptions saa is both tractable and enjoys strong asymptotic performance guarantees similar guarantees however do not typically hold in finite samples in this paper we propose a modification of saa which we term robust saa which retains saa s tractability and asymptotic properties and additionally enjoys strong finite sample performance guarantees the key to our method is linking saa distributionally robust optimization and hypothesis testing of goodness of fit beyond robust saa this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data driven procedures that are based upon distributionally robust optimization this analysis provides insight into the practical performance of these various methods in real applications we present examples from inventory management and portfolio allocation and demonstrate numerically that our approach outperforms other data driven approaches in these applications

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

a cutting surface algorithm for semi infinite convex programming with an application to moment robust optimization we present and analyze a central cutting surface algorithm for general semi infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments moments of arbitrary order as well as nonpolynomial moments can be included in the formulation we show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk aversion with classic robust optimization at one end of the spectrum and stochastic programming at the other although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty the cutting surface method for general semi infinite convex programs is also of independent interest the proposed method is applicable to problems with nondifferentiable semi infinite constraints indexed by an infinite dimensional index set examples comparing the cutting surface algorit

ambiguity in risk preferences in robust stochastic optimization abstract we consider robust stochastic optimization problems for risk averse decision makers where there is ambiguity about both the decision maker s risk preferences and the underlying probability distribution we propose and analyze a robust optimization problem that accounts for both types of ambiguity first we derive a duality theory for this problem class and identify random utility functions as the lagrange multipliers second we turn to the computational aspects of this problem we show how to evaluate our robust optimization problem exactly in some special cases and then we consider some tractable relaxations for the general case finally we apply our model to both the newsvendor and portfolio optimization problems and discuss its implications

expected utility penalty functions and duality in stochastic nonlinear programming we consider nonlinear programming problem p with stochastic constraints the lagrangean corresponding to such problems has a stochastic part which in this work is replaced by its certainty equivalent in the sense of expected utility theory it is shown that the deterministic surrogate problem ce p thus obtained contains a penalty function which penalizes violation of the constraints in the mean the approach is related to several known methods in stochastic programming such as chance constraints stochastic goal programming reliability programming and mean variance analysis the dual problem of ce p is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent nce concept motivation for the nce and its potential role in decision theory are discussed as well as mean variance approximations

a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods