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16
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 86 (15 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
A Hierarchy of NearOptimal Policies for Multistage Adaptive Optimization
, 2009
"... 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 nearoptimal polynomial disturbancefeedback control policies, and show how the ..."
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Cited by 12 (3 self)
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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 nearoptimal polynomial disturbancefeedback 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 tradeoff between the optimality gap and the computational requirements. We evaluate our framework in the context of two classical inventory management applications, in which very strong numerical performance is exhibited, at relatively modest computational expense. 1
Properties of a new parameterization for the control of constrained systems with disturbances
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On the Power and Limitations of Affine Policies in TwoStage Adaptive Optimization
 SUBMITTED TO MATH PROGRAMMING
, 2009
"... We consider a twostage adaptive linear optimization problem under right hand side uncertainty with a minmax 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 p ..."
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Cited by 8 (2 self)
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We consider a twostage adaptive linear optimization problem under right hand side uncertainty with a minmax 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 worstcase cost of an optimal affine policy can be Ω(m 1/2−δ) times the worstcase cost of an optimal fullyadaptable solution for any δ> 0, where m is the number of linear constraints. We also show that the worstcase cost of the best affine policy is O ( √ m) times the optimal cost when the firststage constraint matrix has nonnegative coefficients. Moreover, if there are only k ≤ m uncertain parameters, we generalize the performance bound for affine policies to O ( √ k) which is particularly useful if only a few parameters are uncertain. We also provide an O ( √ 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, U ⊆ R m + is a simplex then an affine policy is optimal. However, an affine policy is suboptimal even if U is a convex combination of only (m + 3) extreme points (only two more extreme points than a simplex) and the worstcase cost of an optimal affine policy can be a factor (2 − δ) worse than the worstcase cost of an optimal fullyadaptable solution for any δ> 0.
Efficient robust optimization for robust control with constraints
 Mathematical Programming, Series A:1–33
, 2007
"... This paper proposes an efficient computational technique for the optimal control of linear discretetime systems subject to bounded disturbances with mixed polytopic constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is n ..."
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Cited by 5 (1 self)
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This paper proposes an efficient computational technique for the optimal control of linear discretetime systems subject to bounded disturbances with mixed polytopic constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is nonconvex. A recent breakthrough has been the application of robust optimization techniques to reparameterise this problem as a convex program. While the reparameterised problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of O(N 6) per iteration of an interiorpoint method. We focus on the case when the disturbance set is ∞norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primaldual interiorpoint methods in which each iteration requires O(N 3) time. This cubic iteration time can be guaranteed using a Riccatibased block factorization technique, which is standard in discretetime optimal control. Numerical results are presented, using a standard sparse primaldual interior point solver, which illustrate the efficiency of this approach.
Offsetfree control of constrained linear discretetime systems subject to persistent unmeasured disturbances
, 2003
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Offsetfree Receding Horizon Control of Constrained Linear Systems subject to Timevarying Setpoints and Persistent Unmeasured Disturbances
, 2003
"... This paper addresses the design of a nonlinear timeinvariant, dynamic state feedback receding horizon controller, which guarantees constraint satisfaction, robust stability and offsetfree control of constrained, linear timeinvariant systems in the presence of timevarying setpoints and unmeasured ..."
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Cited by 4 (0 self)
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This paper addresses the design of a nonlinear timeinvariant, dynamic state feedback receding horizon controller, which guarantees constraint satisfaction, robust stability and offsetfree control of constrained, linear timeinvariant systems in the presence of timevarying setpoints and unmeasured, persistent, additive disturbances. First, this objective is obtained by designing a dynamic, linear timeinvariant, offsetfree controller and an appropriate domain of attraction for this linear controller is defined. The linear (unconstrained) controller is then modified by adding a perturbation term, which is computed by a robust receding horizon controller. It is shown that the domain of attraction of the receding horizon controller contains that of the linear controller and an efficient implementation of the receding horizon controller is proposed. Proofs of robust constraint satisfaction, robust stability and offsetfree control are given. The effectiveness of the proposed controller is illustrated on an example of a continuous stirred tank reactor.
How to match an affine timevarying feedback law: Properties of a new parameterization for the control of constrained systems with disturbances
, 2003
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On robustness of suboptimal minmax model predictive control
 WSEAS TRANSACTIONS on SYSTEMS and CONTROL, Issue 8
, 2007
"... Abstract: With the hard computation of an exact solution to nonconvex optimization problem in a limited time, we propose a suboptimal minmax model predictive control (MPC) scheme for nonlinear discretetime systems subjected to constraints and disturbances. The idea of inputtostate stability (I ..."
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Cited by 2 (0 self)
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Abstract: With the hard computation of an exact solution to nonconvex optimization problem in a limited time, we propose a suboptimal minmax model predictive control (MPC) scheme for nonlinear discretetime systems subjected to constraints and disturbances. The idea of inputtostate stability (ISS) is introduced and a Lyapunovlike sufficient condition for ISS is presented. Based on this, we show that the suboptimal predictive controller obtained here holds back the disturbance robustly in the present of constraints on states and inputs.