Showing papers on "Linear programming published in 2014"

A Model Predictive Control Approach to Microgrid Operation Optimization

Abstract: Microgrids are subsystems of the distribution grid, which comprises generation capacities, storage devices, and controllable loads, operating as a single controllable system either connected or isolated from the utility grid. In this paper, we present a study on applying a model predictive control approach to the problem of efficiently optimizing microgrid operations while satisfying a time-varying request and operation constraints. The overall problem is formulated using mixed-integer linear programming (MILP), which can be solved in an efficient way by using commercial solvers without resorting to complex heuristics or decompositions techniques. Then, the MILP formulation leads to significant improvements in solution quality and computational burden. A case study of a microgrid is employed to assess the performance of the online optimization-based control strategy and the simulation results are discussed. The method is applied to an experimental microgrid located in Athens, Greece. The experimental results show the feasibility and the effectiveness of the proposed approach.

Stability and Stabilization of Linear Systems with Saturating Actuators

Abstract: This monograph details basic concepts and tools fundamental for the analysis and synthesis of linear systems subject to actuator saturation and developments in recent research. The authors use a state-space approach and focus on stability analysis and the synthesis of stabilizing control laws in both local and global contexts. Different methods of modeling the saturation and behavior of the nonlinear closed-loop system are given special attention. Various kinds of Lyapunov functions are considered to present different stability conditions. Results arising from uncertain systems and treating performance in the presence of saturation are given. The text proposes methods and algorithms, based on the use of linear programming and linear matrix inequalities, for computing estimates of the basin of attraction and for designing control systems accounting for the control bounds and the possibility of saturation. They can be easily implemented with mathematical software packages.

Iterative Bregman Projections for Regularized Transportation Problems

Abstract: This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.

Convex Computation of the Region of Attraction of Polynomial Control Systems

Abstract: We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Automatic Security Evaluation and (Related-key) Differential Characteristic Search: Application to SIMON, PRESENT, LBlock, DES(L) and Other Bit-Oriented Block Ciphers

Abstract: We propose two systematic methods to describe the differential property of an S-box with linear inequalities based on logical condition modelling and computational geometry respectively. In one method, inequalities are generated according to some conditional differential properties of the S-box; in the other method, inequalities are extracted from the H-representation of the convex hull of all possible differential patterns of the S-box. For the second method, we develop a greedy algorithm for selecting a given number of inequalities from the convex hull. Using these inequalities combined with Mixed-integer Linear Programming (MILP) technique, we propose an automatic method for evaluating the security of bit-oriented block ciphers against the (related-key) differential attack with several techniques for obtaining tighter security bounds, and a new tool for finding (related-key) differential characteristics automatically for bit-oriented block ciphers.

Security-Constrained Unit Commitment With Linearized System Frequency Limit Constraints

Abstract: Rapidly increasing the penetration level of renewable energies has imposed new challenges to the operation of power systems. Inability or inadequacy of these resources in providing inertial and primary frequency responses is one of the important challenges. In this paper, this issue is addressed within the framework of security-constrained unit commitment (SCUC) by adding new constraints representing the system frequency response. A modified system frequency response model is first derived and used to find analytical representation of system minimum frequency in thermal-dominant multi-machine systems. Then, an effective piecewise linearization (PWL) technique is employed to linearize the nonlinear function representing the minimum system frequency, facilitating its integration in the SCUC problem. The problem is formulated as a mixed-integer linear programming (MILP) problem which is solved efficiently by available commercial solvers. The results indicate that the proposed method can be utilized to integrate renewable resources into power systems without violating system frequency limits.

A Dynamic Near-Optimal Algorithm for Online Linear Programming

Abstract: A natural optimization model that formulates many online resource allocation problems is the online linear programming LP problem in which the constraint matrix is revealed column by column along with the corresponding objective coefficient. In such a model, a decision variable has to be set each time a column is revealed without observing the future inputs, and the goal is to maximize the overall objective function. In this paper, we propose a near-optimal algorithm for this general class of online problems under the assumptions of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Specifically, our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from the revealed columns in the previous period are used to determine the sequential decisions in the current period. Through dynamic learning, the competitiveness of our algorithm improves over the past study of the same problem. We also present a worst case example showing that the performance of our algorithm is near optimal.

Large-Margin Multi-ViewInformation Bottleneck

Abstract: In this paper, we extend the theory of the information bottleneck (IB) to learning from examples represented by multi-view features. We formulate the problem as one of encoding a communication system with multiple senders, each of which represents one view of the data. Based on the precise components filtered out from multiple information sources through a “bottleneck”, a margin maximization approach is then used to strengthen the discrimination of the encoder by improving the code distance within the frame of coding theory. The resulting algorithm therefore inherits all the merits of the IB principle and coding theory. It has two distinct advantages over existing algorithms, namely, that our method finds a tradeoff between the accuracy and complexity of the multi-view model, and that the encoded multi-view data retains sufficient discrimination for classification. We also derive the robustness and generalization error bound of the proposed algorithm, and reveal the specific properties of multi-view learning. First, the complementarity of multi-view features guarantees the robustness of the algorithm. Second, the consensus of multi-view features reduces the empirical Rademacher complexity of the objective function, enhances the accuracy of the solution, and improves the generalization error bound of the algorithm. The resulting objective function is solved efficiently using the alternating direction method. Experimental results on annotation, classification and recognition tasks demonstrate that the proposed algorithm is promising for practical applications.

Optimal Geo-Indistinguishable Mechanisms for Location Privacy

Abstract: We consider the geo-indistinguishability approach to location privacy, and the trade-off with respect to utility. We show that, given a desired degree ofgeo-indistinguishability, it is possible to construct a mechanism that minimizes the service quality loss, using linear programming techniques. In addition we show that, under certain conditions, such mechanism also provides optimal privacy in the sense of Shokri et al. Furthermore, we propose a method to reduce the number of constraints of the linear program from cubic to quadratic, maintaining the privacy guarantees and without affecting significantly the utility of the generated mechanism. This reduces considerably the time required to solve the linear program, thus enlarging significantly the location sets for which the optimal mechanisms can be computed.

Optimal Geo-Indistinguishable Mechanisms for Location Privacy

Abstract: We consider the geo-indistinguishability approach to location privacy, and the trade-off with respect to utility. We show that, given a desired degree of geo-indistinguishability, it is possible to construct a mechanism that minimizes the service quality loss, using linear programming techniques. In addition we show that, under certain conditions, such mechanism also provides optimal privacy in the sense of Shokri et al. Furthermore, we propose a method to reduce the number of constraints of the linear program from cubic to quadratic, maintaining the privacy guarantees and without affecting significantly the utility of the generated mechanism. This reduces considerably the time required to solve the linear program, thus enlarging significantly the location sets for which the optimal mechanisms can be computed.

An $O(1/k)$ Gradient Method for Network Resource Allocation Problems

Abstract: We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).

Energy-Efficient Stochastic Task Scheduling on Heterogeneous Computing Systems

Abstract: In the past few years, with the rapid development of heterogeneous computing systems (HCS), the issue of energy consumption has attracted a great deal of attention. How to reduce energy consumption is currently a critical issue in designing HCS. In response to this challenge, many energy-aware scheduling algorithms have been developed primarily using the dynamic voltage-frequency scaling (DVFS) capability which has been incorporated into recent commodity processors. However, these techniques are unsatisfactory in minimizing both schedule length and energy consumption. Furthermore, most algorithms schedule tasks according to their average-case execution times and do not consider task execution times with probability distributions in the real-world. In realizing this, we study the problem of scheduling a bag-of-tasks (BoT) application, made of a collection of independent stochastic tasks with normal distributions of task execution times, on a heterogeneous platform with deadline and energy consumption budget constraints. We build execution time and energy consumption models for stochastic tasks on a single processor. We derive the expected value and variance of schedule length on HCS by Clark's equations. We formulate our stochastic task scheduling problem as a linear programming problem, in which we maximize the weighted probability of combined schedule length and energy consumption metric under deadline and energy consumption budget constraints. We propose a heuristic energy-aware stochastic task scheduling algorithm called ESTS to solve this problem. Our algorithm can achieve high scheduling performance for BoT applications with low time complexity $O(n(M+\log n))$ , where $n$ is the number of tasks and $M$ is the total number of processor frequencies. Our extensive simulations for performance evaluation based on randomly generated stochastic applications and real-world applications clearly demonstrate that our proposed heuristic algorithm can improve the weighted probability that both the deadline and the energy consumption budget constraints can be met, and has the capability of balancing between schedule length and energy consumption.

Optimal Power Flow Using Particle Swarm Optimization

Abstract: 2 Abstract: This paper proposes the application of Particle Swarm Optimization (PSO) technique to solve Optimal Power Flow with inequality constraints on Line Flow. To ensure secured operation of power system, it i s necessary to keep the line flow within the prescribed MVA limit so that the system operates in normal state. The problem involves non-linear objective function and constraints. Therefore, the population based method like PSO is more suitable than the conventional Linear Programming methods. This approach is applied to a six bus three unit system and the results are compared with results of Linear Programming method for different test cases. The obtained solution proves that the proposed technique is efficient and accurate.

Learning to Search in Branch and Bound Algorithms

Abstract: Branch-and-bound is a widely used method in combinatorial optimization, including mixed integer programming, structured prediction and MAP inference. While most work has been focused on developing problem-specific techniques, little is known about how to systematically design the node searching strategy on a branch-and-bound tree. We address the key challenge of learning an adaptive node searching order for any class of problem solvable by branch-and-bound. Our strategies are learned by imitation learning. We apply our algorithm to linear programming based branch-and-bound for solving mixed integer programs (MIP). We compare our method with one of the fastest open-source solvers, SCIP; and a very efficient commercial solver, Gurobi. We demonstrate that our approach achieves better solutions faster on four MIP libraries.

Algorithms for Leader Selection in Stochastically Forced Consensus Networks

Abstract: We are interested in assigning a pre-specified number of nodes as leaders in order to minimize the mean-square deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (which indicate whether a node is a leader) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm well-suited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms.

Optimal Decisions from Probabilistic Models: The Intersection-over-Union Case

Abstract: A probabilistic model allows us to reason about the world and make statistically optimal decisions using Bayesian decision theory. However, in practice the intractability of the decision problem forces us to adopt simplistic loss functions such as the 0/1 loss or Hamming loss and as result we make poor decisions through MAP estimates or through low-order marginal statistics. In this work we investigate optimal decision making for more realistic loss functions. Specifically we consider the popular intersection-over-union (IoU) score used in image segmentation benchmarks and show that it results in a hard combinatorial decision problem. To make this problem tractable we propose a statistical approximation to the objective function, as well as an approximate algorithm based on parametric linear programming. We apply the algorithm on three benchmark datasets and obtain improved intersection-over-union scores compared to maximum-posterior-marginal decisions. Our work points out the difficulties of using realistic loss functions with probabilistic computer vision models.

Optimal Supervisory Control of Flexible Manufacturing Systems by Petri Nets: A Set Classification Approach

Abstract: Supervisory control is usually considered as an external control mechanism to a system by controlling the occurrences of its controllable events. There exist Petri net models whose legal reachability spaces are nonconvex. In this case, they cannot be optimally controlled by the conjunctions of linear constraints. For Petri net models of flexible manufacturing systems, this work proposes a method to classify the legal markings into several subsets. Each subset is associated with a linear constraint that can forbid all first-met bad markings. Then, the disjunctions of the obtained constraints can make all legal markings reachable and forbid all first-met bad markings, i.e., the controlled net is live and maximally permissive. An integer linear programming model is formulated to minimize the number of the constraints. A supervisory structure is also proposed to implement the disjunctions of the constraints. Finally, examples are provided to illustrate the proposed method.

The matching polytope has exponential extension complexity

Abstract: A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher dimensional polytope Q that can be linearly projected on P. However, the central question in this field remained wide open: can the perfect matching polytope be written as an LP with polynomially many constraints? We answer this question negatively. In fact, the extension complexity of the perfect matching polytope in a complete n-node graph is 2Ω(n). By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω(√n) to 2Ω(n).

Convex optimization of nonlinear feedback controllers via occupation measures

Abstract: The construction of feedback control laws for underactuated nonlinear robotic systems with input saturation limits is crucial for dynamic robotic tasks such as walking, running, or flying. Existing techniques for feedback control design are either restricted to linear systems, rely on discretizations of the state space, or require solving a nonconvex optimization problem that requires feasible initialization. This paper presents a method for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). In contrast to traditional approaches based on Lyapunov's criteria for stability, we rely on the notion of occupation measures to pose this problem as an infinite-dimensional linear program which can then be approximated in finite dimension via semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS which is well suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on six nonlinear systems. Comparisons to an infinite-horizon linear quadratic regulator approach and an approach relying on Lyapunov's criteria for stability are also included in order to illustrate the improved performance of the presented technique.

Convex computation of the maximum controlled invariant set for polynomial control systems

Abstract: We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

Faster Parameterized Algorithms Using Linear Programming

Abstract: We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an Oa(2.618k) algorithm for the problem. Here, k is the excess of the vertex cover size over the LP optimum, and we write Oa(f(k)) for a time complexity of the form O(f(k)nO(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of Oa(2.3146k). Following this, using previously known as well as new reductions, we give Oa(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, and Almost 2-SAT, and Oa(1.5214k) algorithms for Konig Vertex Deletion and Vertex Cover parameterized by the size of the smallest odd cycle transversal and Konig vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal—this is the first algorithm that improves on the dependence on k of the seminal Oa(3k) algorithm of Reed, Smith, and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k − clog k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.