TY - JOUR A1 - Guigues, Vincent A1 - Henrion, Rene T1 - Joint dynamic probabilistic constraints with projected linear decision rules JF - Optimization Methods and Software N2 - We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically. Y1 - 2016 U6 - https://doi.org/10.1080/10556788.2016.1233972 VL - 32 SP - 1006 EP - 1032 ER - TY - JOUR A1 - Diniz, Andre Luiz A1 - Henrion, Rene T1 - On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions JF - Energy Systems N2 - Many engineering problems with uncertain data, notably arising in power management, can be formulated as optimization problems subject to probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task in general both from the numerical and theoretical point of view, quite some progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set of feasible decisions. Y1 - 2016 U6 - https://doi.org/10.1007/s12667-015-0180-6 VL - 8 SP - 149 EP - 167 ER - TY - JOUR A1 - Gugat, Martin A1 - Zuazua, Enrique ED - Grimble, Mike J. T1 - Exact penalization of terminal constraints for optimal control problems JF - OPTIMAL CONTROL APPLICATIONS AND METHODS N2 - We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd. Y1 - 2016 U6 - https://doi.org/10.1002/oca.2238 VL - 37 IS - 6 SP - 1329 EP - 1354 ER - TY - CHAP A1 - Hante, Falk T1 - On the relaxation gap for PDE mixed-integer optimal control problems T2 - Proceedings in Applied Mathematics and Mechanics Y1 - 2016 U6 - https://doi.org/10.1002/pamm.201610380 VL - 16 SP - 783 EP - 784 ER - TY - JOUR A1 - Gugat, Martin A1 - Trelat, Emmanuel A1 - Zuazua, Enrique ED - Sepulchre, Rodolphe T1 - Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property JF - Systems & Control Letters N2 - We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time TT, by minimizing an objective functional that is the convex sum of the L2L2-norm of the control and of a boundary Neumann tracking term. We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time TT is large, then the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 00. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If T=+∞T=+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0,T][0,T] and coincides with the control given by the Hilbert Uniqueness Method. In addition, we establish a similarity theorem stating that, for every T>0T>0, there exists an appropriate weight λ<1λ<1 for which the optimal solutions of the corresponding finite horizon optimal control problem and of the infinite horizon optimal control problem coincide along the first part of the time interval [0,2][0,2]. We also discuss the turnpike phenomenon from the perspective of a general framework with a strongly continuous semi-group. KW - Neumann boundary control KW - Turnpike phenomenon KW - Exact control Y1 - 2016 U6 - https://doi.org/10.1016/j.sysconle.2016.02.001 VL - 90 SP - 61 EP - 70 ER - TY - JOUR A1 - Grimm, Veronika A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weibelzahl, Martin A1 - Zöttl, Gregor T1 - Transmission and generation investment in electricity markets: The effects of market splitting and network fee regimes JF - European Journal of Operational Research N2 - We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes. KW - Electricity market modeling KW - Mixed-integer nonlinear optimization KW - Multilevel programming KW - Network expansion KW - Transmission management Y1 - 2016 U6 - https://doi.org/10.1016/j.ejor.2016.03.044 VL - 254 IS - 2 SP - 493 EP - 509 ER - TY - JOUR A1 - Hante, Falk A1 - Mommer, Mario A1 - Potschka, Andreas T1 - Newton-Picard preconditioners for time-periodic, parabolic optimal control problems JF - SIAM Journal on Numerical Analysis Y1 - 2016 U6 - https://doi.org/10.1137/140967969 VL - 53 IS - 5 SP - 2206 EP - 2225 ER - TY - JOUR A1 - Liers, Frauke A1 - Merkert, Maximilian T1 - Structural Investigation of Piecewise Linearized Network Flow Problems N2 - In this work we study polyhedra in the context of network flow problems, where the flow value on each arc lies in one of several predefined intervals. This is motivated by nonlinear problems on transportation networks, where nonlinearities are handled by piecewise linear approximation or relaxation - a common and established approach in many applications. Several methods for modeling piecewise linear functions are known which provide a complete description for a single network arc. However, in general this property is lost when considering multiple arcs. We show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities. For the case of paths of degree-two-nodes we give a complete description of the polyhedron projected to the integer variables. Our model is based on - but not limited to - the multiple choice method; we also show how to transfer our results to a formulation based on the incremental method. Computational results show that a state-of-the-art MIP-solver greatly benefits from using our cutting planes for random and realistic network topologies. KW - Combinatorial optimization KW - Complete description KW - Network flow problems KW - Piecewise linear functions Y1 - 2016 U6 - https://doi.org/10.1137/15M1006751 VL - 26 SP - 2863 EP - 2886 ER - TY - JOUR A1 - Cacchiani, Valentina A1 - Jünger, Michael A1 - Liers, Frauke A1 - Lodi, Andrea A1 - Schmidt, Daniel T1 - Single-commodity robust network design with finite and Hose demand sets JF - Mathematical Programming N2 - We study a single-commodity robust network design problem (sRND) defined on an undirected graph. Our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single-commodity flow. We consider two ways of representing the uncertainty set, either as a finite list of scenarios or as a polytope. We propose a branch-and-cut algorithm to derive optimal solutions to sRND, built on a capacity-based integer linear programming formulation. It is strengthened with valid inequalities derived as {0, 1/2}-Chvátal–Gomory cuts. Since the formulation contains exponentially many constraints, we provide practical separation algorithms. Extensive computational experiments show that our approach is effective, in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver. KW - Robust network design KW - Branch-and-cut KW - Cut-set inequalities KW - Polyhedral demand uncertainty KW - Separation under uncertainty Y1 - 2016 U6 - https://doi.org/10.1007/s10107-016-0991-9 VL - 157 IS - 1 SP - 297 EP - 342 ER - TY - JOUR A1 - Pfaff, Sebastian A1 - Ulbrich, Stefan T1 - Optimal Control of Nonlinear Hyperbolic Conservation Laws by On/Off-Switching JF - Optimization Methods and Software N2 - This paper studies the differentiability properties of the control-to-state mapping for entropy solutions to a scalar hyperbolic conservation law on R with respect to the switching times of an on/off-control. The switching times between on-modes and off-modes are the control variables of the considered optimization problem, where a general tracking-type functional is minimized.We investigate the differentiability of the reduced objective function, also in the presence of shocks. We show that the state y(t,·) at some observation time t depends differentiably on the switching times in a generalized sense that implies total differentiability for the composition with a tracking functional. Furthermore, we present an adjoint-based formula for the gradient of the reduced objective functional with respect to the switching times. KW - optimal control, scalar conservation law, network Y1 - 2017 U6 - https://doi.org/10.1080/10556788.2016.1236796 VL - 32 SP - 904 EP - 939 ER - TY - JOUR A1 - Pfaff, Sebastian A1 - Ulbrich, Stefan T1 - Optimal Boundary Control of Nonlinear Hyperbolic Conservation Laws with Switched Boundary Data JF - SIAM Journal on Control and Optimization N2 - We consider the optimal control of initial-boundary value problems for entropy solutions of scalar hyperbolic conservation laws. In particular, we consider initial-boundary value problems where the initial and boundary data switch between different C¹-functions at certain switching points and both the functions and the switching points are controlled. We show that the control-to-state mapping is differentiable in a certain generalized sense, which implies Fréchet-differentiability with respect to the control functions and the switching points for the composition with a tracking type functional, even in the presence of shocks. We also present an adjoint-based formula for the gradient of the reduced objective functional. KW - optimal control, scalar conservation law, differentiability, adjoint state, shock sensitivity Y1 - 2016 U6 - https://doi.org/10.1137/140995799 VL - 53 IS - 3 SP - 1250 EP - 1277 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Solving Power-Constrained Gas Transportation Problems using an MIP-based Alternating Direction Method JF - Computers & Chemical Engineering N2 - We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to difficult nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the nonlinearities and a tailored alternating direction method. Most other publications in the field of gas transport optimization only consider pressure and flow as main physical quantities. In this work, we additionally incorporate heat power supplies and demands as well as a mixing model for different gas qualities. We demonstrate the capabilities of our method on Germany's largest transport networks and hereby present numerical results on the largest instances that were ever reported in the literature for this problem class. Y1 - 2016 U6 - https://doi.org/10.1016/j.compchemeng.2015.07.005 VL - 82 IS - 2 SP - 303 EP - 317 ER - TY - JOUR A1 - Rüffler, Fabian A1 - Hante, Falk T1 - Optimal Switching for Hybrid Semilinear Evolutions JF - Nonlinear Analysis: Hybrid Systems Y1 - 2016 U6 - https://doi.org/10.1016/j.nahs.2016.05.001 VL - 22 SP - 215 EP - 227 ER -