adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

high accuracy non polynomial spline in compression method for one space dimensional quasi linear hyperbolic equations with significant first order space derivative term abstract in this paper we propose a new three level implicit nine point compact non polynomial spline in compression finite difference method of order two in time and four in space directions based on non polynomial spline approximation in x direction and central difference approximation in t direction for the numerical solution of one space dimensional second order quasi linear hyperbolic partial differential equations with first order space derivative term we describe the mathematical details of the method and also discuss how our method is able to handle wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable numerical results are provided to justify the usefulness of the proposed method

on the performance of affine policies for two stage adaptive optimization a geometric perspective we consider two stage adjustable robust linear optimization problems with uncertain right hand side mathbf b b belonging to a convex and compact uncertainty set mathcal u u we provide an a priori approximation bound on the ratio of the optimal affine in mathbf b b solution to the optimal adjustable solution that depends on two fundamental geometric properties of mathcal u u a the symmetry and b the simplex dilation factor of the uncertainty set mathcal u u and provides deeper insight on the power of affine policies for this class of problems the bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature we also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem illustrating that the proposed bound is informative

a new unconditionally stable compact difference scheme of o τ2 h4 for the 1d linear hyperbolic equation abstract in this paper we propose a three level compact difference scheme of o 2 h 4 for the difference solution of one dimensional second order non homogeneous linear hyperbolic equation u tt 2 u t 2 u u xx f x t 0 x 1 t 0 where 0 stability analysis of the method has been carried out finally numerical examples are used to illustrate the efficiency of the new difference scheme

multistage robust mixed integer optimization with adaptive partitions we present a new partition and bound method for multistage adaptive mixed integer optimization amio problems that extends previous work on finite adaptability the approach analyzes the optimal solution to a static nonadaptive version of an amio problem to gain insight into which regions of the uncertainty set are restricting the objective function value we use this information to construct partitions in the uncertainty set leading to a finitely adaptable formulation of the problem we use the same information to determine a lower bound on the fully adaptive solution the method repeats this process iteratively to further improve the objective until a desired gap is reached we provide theoretical motivation for this method and characterize its convergence properties and the growth in the number of partitions using these insights we propose and evaluate enhancements to the method such as warm starts and smarter partition creation we describe in detail how to apply finite adaptability to multista

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

piecewise static policies for two stage adjustable robust linear optimization in this paper we consider two stage adjustable robust linear optimization problems under uncertain constraints and study the performance of piecewise static policies these are a generalization of static policies where we divide the uncertainty set into several pieces and specify a static solution for each piece we show that in the worst case there is no piecewise static policy with a polynomial number of pieces that has a significantly better performance than an optimal static policy this is quite surprising as piecewise static policies are significantly more general than static policies the proof is based on a combinatorial argument and the structure of piecewise static policies

a nonlinear semidefinite optimization relaxation for the worst case linear optimization under uncertainties in this paper we consider the so called worst case linear optimization wclo with uncertainties in the right hand side of the constraints such a problem arises from numerous applications such as systemic risk estimate in finance and stochastic optimization we first show the problem is np hard and present a coarse semidefinite relaxation sdr for wclo an iterative procedure is introduced to sequentially refine the relaxation model based on the solution of the current relaxation model by simply changing some parameters in the coarse sdr we show that the sequence of the proposed sdrs will converge to some nonlinear semidefinite optimization problem sdo a bi section search algorithm is proposed to solve the resulting nonlinear sdo our preliminary experimental results illustrate that the nonlinear sdr can provide very tight bounds for the original wclo and is able to locate the exact global solution in most cases within a few iterations on average

design of near optimal decision rules in multistage adaptive mixed integer optimization in recent years decision rules have been established as the preferred solution method for addressing computationally demanding multistage adaptive optimization problems despite their success existing decision rules a are typically constrained by their a priori design and b do not incorporate in their modeling adaptive binary decisions to address these problems we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions respectively we then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed integer optimization we demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems we provide global lower bounds and show that our approach is i practically tractable and ii provides high quality solutions that outperform alternative methods

design of affine controllers via convex optimization we consider a discrete time time varying linear dynamical system perturbed by process noise with linear noise corrupted measurements over a finite horizon we address the problem of designing a general affine causal controller in which the control input is an affine function of all previous measurements in order to minimize a convex objective in either a stochastic or worst case setting this controller design problem is not convex in its natural form but can be transformed to an equivalent convex optimization problem by a nonlinear change of variables which allows us to efficiently solve the problem our method is related to the classical design procedure for time invariant infinite horizon linear controller design and the more recent purified output control method we illustrate the method with applications to supply chain optimization and dynamic portfolio optimization and show the method can be combined with model predictive control techniques when perfect state information is available

extending scope of robust optimization comprehensive robust counterparts of uncertain problems in this paper we propose a new methodology for handling optimization problems with uncertain data with the usual robust optimization paradigm one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set whereas with the proposed approach we require also a controlled deterioration in performance when the data is outside the uncertainty set the extension of robust optimization methodology developed in this paper opens up new possibilities to solve efficiently multi stage finite horizon uncertain optimization problems in particular to analyze and to synthesize linear controllers for discrete time dynamical systems

optimality of affine policies in multistage robust optimization in this paper we prove the optimality of disturbance affine control policies in the context of one dimensional constrained multistage robust optimization our results cover the finite horizon case with minimax worst case objective and convex state costs plus linear control costs we develop a new proof methodology which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function apart from providing an elegant and conceptually simple proof technique the approach also entails very fast algorithms for the case of piecewise affine state costs which we explore in connection with a classical inventory management application

on the adaptivity gap in two stage robust linear optimization under uncertain packing constraints in this paper we study the performance of static solutions in two stage adjustable robust packing linear optimization problem with uncertain constraint coefficients such problems arise in many important applications such as revenue management and resource allocation problems where demand requests have uncertain resource requirements the goal is to find a two stage solution that maximizes the worst case objective value over all possible realizations of the second stage constraints from a given uncertainty set we consider the case where the uncertainty set is column wise and constraint wise any constraint describing the set involve entries of only a single column or a single row this is a fairly general class of uncertainty sets to model constraint coefficient uncertainty we show that the two stage adjustable robust problem is varomega log n hard to approximate on the positive side we show that a static solution is an o big log n cdot min log vargamma log m n big approximation for the two stage adjustable robust problem where m and n denote the numbers of rows and columns of the constraint matrix and vargamma is the maximum possible ratio of upper bounds of the uncertain constraint coefficients therefore for constant vargamma surprisingly the performance bound for static solutions and therefore the adaptivity gap matches the hardness of approximation for the adjustable problems furthermore in general the static solution provides nearly the best efficient approximation for the two stage adjustable robust problem

a new high order space derivative discretization for 3d quasi linear hyperbolic partial differential equations abstract in this paper we propose a new high accuracy numerical method of o k2 k2h2 h4 for the solution of three dimensional quasi linear hyperbolic partial differential equations where k 0 and h 0 are mesh sizes in time and space directions respectively we mainly discretize the space derivative terms using fourth order approximation and time derivative term using second order approximation we describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable the proposed method behaves like a fourth order method for a fixed value of k h 2 some examples and their numerical results are provided to justify the usefulness of the proposed method

infinite horizon performance bounds for uncertain constrained systems we present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances we take as a performance metric either an expected value or minimax discounted cost over an infinite horizon and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances and we show that the two bounds are closely related in particular the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set we show that our performance bounds can be computed via solution of a finite dimensional convex optimization problem and provide numerical examples to illustrate the efficacy of our method

on the power and limitations of affine policies in two stage adaptive optimization we consider a two stage adaptive linear optimization problem under right hand side uncertainty with a min max objective and give a sharp characterization of the power and limitations of affine policies where the second stage solution is an affine function of the right hand side uncertainty in particular we show that the worst case cost of an optimal affine policy can be omega m 1 2 delta times the worst case cost of an optimal fully adaptable solution for any 0 where m is the number of linear constraints we also show that the worst case cost of the best affine policy is o sqrt m times the optimal cost when the first stage constraint matrix has non negative coefficients moreover if there are only k m uncertain parameters we generalize the performance bound for affine policies to o sqrt k which is particularly useful if only a few parameters are uncertain we also provide an o sqrt k approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters we also give a tight characterization of the conditions under which an affine policy is optimal for the above model in particular we show that if the uncertainty set mathcal u subseteq mathbb r m is a simplex then an affine policy is optimal however an affine policy is suboptimal even if mathcal u is a convex combination of only m 3 extreme points only two more extreme points than a simplex and the worst case cost of an optimal affine policy can be a factor 2 worse than the worst case cost of an optimal fully adaptable solution for any 0

a numerical algorithm for the solution of telegraph equations abstract in this paper we present a new competitive numerical scheme to solve nonlinear telegraph equations the method is based on rothe s approximation in time discretization and on the wavelet galerkin in the spatial discretization the approximate solutions converge in the space c 0 t l 2 l 2 0 t w 0 1 2 to the variational solution a full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

when are static and adjustable robust optimization problems with constraint wise uncertainty equivalent adjustable robust optimization aro generally produces better worst case solutions than static robust optimization ro however aro is computationally more difficult than ro in this paper we provide conditions under which the worst case objective values of aro and ro problems are equal we prove that when the uncertainty is constraint wise the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact then robust solutions are also optimal for the corresponding aro problem furthermore we prove that if some of the uncertain parameters are constraint wise and the rest are not then under a similar set of assumptions there is an optimal decision rule for the aro problem that does not depend on the constraint wise uncertain parameters also we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non constraint wise uncertain parameters finally we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems

a new fast algorithm based on half step discretization for one space dimensional quasilinear hyperbolic equations abstract in this article we describe a new compact three level implicit method of order four in time and space based on half step discretization for one space dimensional quasilinear hyperbolic equation u tt a x t u u xx f x t u u x u t defined in the semi infinite solution region where a 0 we require only nine grid points for the unknown variable u x t and two extra half step points each for x and t variables the proposed method is directly applicable to wave equation with singular coefficients which is main attraction of our work we do not require extra grid points for computation we describe the derivation of the method in detail the proposed method when applied to damped wave equation is shown to be unconditionally stable many benchmark problems are solved to confirm the fourth order convergence of the proposed method

on the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients abstract implicit difference schemes of o k 4 k 2 h 2 h 4 where k 0 h 0 are grid sizes in time and space coordinates respectively are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and dirichlet boundary conditions the proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry the numerical examples are given to illustrate the fourth order convergence of the methods

on the power of robust solutions in two stage stochastic and adaptive optimization problems we consider a two stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution in particular we show that if the right hand side of the constraints is uncertain and belongs to a symmetric uncertainty set such as hypercube ellipsoid or norm ball and the probability measure is also symmetric then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two stage stochastic problem furthermore we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric we refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two stage stochastic problem as the stochasticity gap we also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive if both the objective coefficients and right hand side are uncertain we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric however we prove that the adaptability gap ratio of optimal cost of the robust problem and the optimal cost of a two stage fully adaptable problem is at most four even if both the objective coefficients and the right hand side of the constraints are uncertain and belong to a symmetric uncertainty set the bound holds for the class of positive uncertainty sets as well moreover if the uncertainty set is a hypercube special case of a symmetric set the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain

characteristics of mixed traffic flow with non motorized vehicles and motorized vehicles at an unsignalized intersection abstract in this paper a new two dimensional car following model is proposed to depict the features of mixed traffic flow consisting of motorized vehicles m vehicle and non motorized vehicles n m vehicle based on the two dimensional optimal velocity ov model by nakayama et al a nakayama k hasebe y sugiyama phys rev e 71 2005 036121 in the proposed model velocity difference terms are introduced which are regarded as important factors for traffic behavior numerical simulations are carried out to investigate the interaction between left turning n m vehicle flow and straight going m vehicle flow at a typical unsignalized interaction the results show that the straight going m vehicle flow just next to n m lane is disturbed more seriously than others in addition a well known phenomenon in reality is observed that groups of m vehicles and n m vehicles pass through the intersection alternately

selected topics in robust convex optimization robust optimization is a rapidly developing methodology for handling optimization problems affected by non stochastic uncertain but bounded data perturbations in this paper we overview several selected topics in this popular area specifically 1 recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data 2 tractability of robust counterparts 3 links between ro and traditional chance constrained settings of problems with stochastic data and 4 a novel generic application of the ro methodology in robust linear control

primal and dual linear decision rules in stochastic and robust optimization linear stochastic programming provides a flexible toolbox for analyzing real life decision situations but it can become computationally cumbersome when recourse decisions are involved the latter are usually modeled as decision rules i e functions of the uncertain problem data it has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence in this paper we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program by employing techniques that are commonly used in modern robust optimization we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes the gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation our method remains applicable if the stochastic program has random recourse and multiple decision stages it also extends to cases involving ambiguous probability distributions

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed