Abstract—In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1,-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory. Index Terms—Hybrid systems, Lyapunov stability, model predictive control (MPC), piecewise affine systems. I.

This paper describes ASTA, an Artificial Stock Trading Agent, in the Matlab programming environment. The primary purpose of the project is to supply a stable and realistic test bench for the development of multi-stock trading algorithms. The behavior of the agent is controlled by a high-level language, which is easily extendable with user-defined functions. The buy and sell rules can be composed interactively and various types of data screening can be easily performed, all within the Matlab m-file language syntax. Apart from

The paper presents a Graphical User Interface (GUI) for nonlinear programming in Matlab. The GUI gives easy access to all features in the NLPLIB TB (NonLinear Programming LI-Brary Toolbox) � a set of Matlab solvers, test problems, graphical and computational utilities for unconstrained and constrained optimization, quadratic programming, unconstrained and constrained nonlinear least squares, box-bounded global optimization, global mixed-integer nonlinear programming, and exponential sum model tting. The GUI also runs the linear programming problems in the linear and discrete optimization toolbox OPERA TB. Both NLPLIB TB and OPERA TB are part of TOMLAB � an environment in Matlab for research and teaching in optimization. Presently, NLPLIB TB implements more than 25 solver algorithms, and it is possible to call solvers in the Math Works Optimization Toolbox. MEX- le interfaces are developed for seven Fortran and C solvers, and others are easily added using the same type of interface routines. There are four ways to solve a problem: by a direct call to the solver routine or a call to amulti-solver driver routine, or interactively, using the Graphical User Interface or a menu system. The GUI may alsobe used as a preprocessor to generate Matlab code for stand-alone runs. Alargeset of standard test problems is implemented in TOMLAB. Furthermore, using MEX- le interfaces, problems in the CUTE test problem data base and problems de ned in the AMPL modeling language can be solved.

We describe a method to find low cost employee shift schedules that guarantee that the fraction of customers who wait less than a specified time (the service level) is always at or above a specified minimum. Most previous approaches used a two-step procedure: (1) determine period-by-period staffing requirements, and (2) find a minimum cost schedule that provides the required number of employees in every period. Due to approximations used in the first step, the two-step approach sometimes results in infeasible or suboptimal solutions. Our method iterates between a schedule evaluator and a schedule generator. Each iteration begins with a schedule for which the schedule evaluator calculates transient service levels using the randomization method. The transient service levels are used to identify infeasible intervals, where the service level is lower than desired. The schedule generator solves a series of integer programs to produce improved schedules. One constraint is added to the integer program for every infeasible interval, in an attempt to eliminate infeasibility without eliminating the optimal solution. The procedure terminates when a feasible solution is found. We present results for a range of test problems and discuss factors that make our approach more likely to outperform previous approaches. 1.

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The Matlab toolbox OPERA TB is a set of Matlab m- les, which solves basic linear and discrete optimization problems in operations research and mathematical programming. Included are routines for linear programming (LP), network programming (NP), integer programming (IP) and dynamic programming (DP). OPERA TB, like the nonlinear programming toolbox NLPLIB TB, is a part of TOMLAB � an environment in Matlab for research and teaching in optimization. Linear programs are solved either by direct call to a solver routine or to a multisolver driver routine, or interactively, using the Graphical User Interface (GUI) or a menu system. From OPERA TB it is possible to call solvers in the Math Works Optimization Toolbox and, using a MEX- le interface, general-purpose solvers implemented in Fortran or C. The focus is on dense problems, but sparse linear programs may be solved using the commercial solver MINOS. Presently, OPERA TB implements about thirty algorithms and includes a set of test examples and demonstration les. This paper gives an overview of OPERA TB and presents test results for medium size LP problems. The tests show that the OPERA TB solver converges as fast as commercial Fortran solvers and is at least ve times faster than the simplex LP solver in the Optimization Toolbox 2.0andtwice as fast as the primal-dual interior-pointLP solver in the same toolbox. Running the commercial Fortran solvers using MEX- le interfaces gives a speed-up factor of ve to thirty- ve.

In most enterprises, databases are deployed on dedicated database servers. Often, these servers are underutilized much of the time. For example, in traces from almost 200 production servers from different organizations, we see an average CPU utilization of less than 4%. This unused capacity can be potentially harnessed to consolidate multiple databases on fewer machines, reducing hardware and operational costs. Virtual machine (VM) technology is one popular way to approach this problem. However, as we demonstrate in this paper, VMs fail to adequately support database consolidation, because databases place a unique and challenging set of demands on hardware resources, which are not well-suited to the assumptions made by VM-based consolidation. Instead, our system for database consolidation, named Kairos, uses novel techniques to measure the hardware requirements of database workloads, as well as models to predict the combined resource utilization of those workloads. We formalize the consolidation problem as a non-linear optimization program, aiming to minimize the number of servers and balance load, while achieving near-zero performance degradation. We compare Kairos against virtual machines, showing up to a factor of 12 × higher throughput on a TPC-C-like benchmark. We also tested the effectiveness of our approach on real-world data collected from production servers

Contents 1 The TOMLAB OpBP-yD-fP0 Environment 7 1.1 Backgrou nd................................................. 8 1.2 TheDesignofTOMLAB.......................................... 8 1.2.1 Global Variables . . ........................................ 22 1.3 SolverRou tinesinTOMLAB ....................................... 24 1.4 Menu ProgramsinTOMLAB....................................... 26 1.5 LowLevelRou tinesandTestProblems.................................. 27 1.5.1 Utility Test RouNEFF ........................................ 34 1.6 TheGraphicalUserInterface ....................................... 34 1.6.1 The Advanced Mode ........................................ 35 2 Solving Linear, Quadratic and Integer Programming Problems 39 2.1 Linear Programming Problems ...................................... 39 2.1.1 AQu0 k Linear Programming SoluxFz .............................. 39 2.1.2 Several Linear Programs ..................................... 39 2.1.

In this paper we will discuss the efficiency and implementation details of an algorithm for finding the global minimum of a multivariate function subject to simple bounds on the variables. The algorithm, DIRECT, developed by D. R. Jones, C. D. Perttunen and B. E. Stuckman is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. We have implemented the DIRECT algorithm in Matlab and the efficiency of our implementation is analyzed by comparing it to the result of Jones's implementation on nine standard test problems for global optimization. In fifteen out of eighteen runs the results is to the favor of our implementation. For some test problems the differences in the number of function evaluations needed for the algorithm to converge are small but for others the differences are great enough to be worth a discussion. Our code is integrated in the NLPLIB TB Toolbox as part of the optimization environment TOMLAB. All tests are perfor...

TOMLAB is a Matlab 5 development environment for research and teaching in optimization, running on both Unix and PCsystems. One motivation for TOMLAB is to simplify research on practical optimization problems, giving easy access to all types of solvers; at the same time having full access to the power of Matlab. In this paper we discuss the design and contents of TOMLAB, as well as some applications where TOMLAB has been successfully applied. TOMLAB includes routines for linear, integer and nonlinear programming, nonlinear parameter estimation and global optimization. More than 65 algorithms are implemented, together with graphical and computational utilities, menu systems and a graphical user interface. It is possible to call solvers in the Math Works Optimization Toolbox and, using MEX-file interfaces, general-purpose solvers implemented in Fortran or C. TOMLAB implements powerful and robust state-of-the-art routines for nonlinear parameter estimation and global optimization; areas of special interest in applied research. Results on practical applications in these areas will be discussed. Our new algorithms for constrained nonlinear least squares algorithms are out-performing the state-of-the-art commercial solvers on classical test problems. For exponential sum model fitting on real-life data from radiotherapy planning our new separable nonlinear least squares algorithm is converging> 35 % faster than other solvers. 1