Standard Simplex Research Articles

Model formulation and solution algorithm of traffic signal control in an urban network

The existing network traffic signal optimization formulations usually do not include traffic flow models, except for control schemes such as SCOOT (Split, Cycle, and Offset Optimization Technique) system that uses simulation for heuristic optimization. Other conventional models normally use isolated intersection optimization with traffic arrival prediction using detector information, or optimization schemes based on green bandwidth approach such as MAXBAND (Maximal Bandwidth). In this paper we present a complete formulation of the problem that includes explicit constraints to model the movement of traffic along the streets between the intersections in a time-expanded network, as well as constraints to capture the permitted movements from modern signal controllers. The platoon dispersion model used is the well-known Robertson's model, which forms linear constraints. Thus it is a rare example of a traffic simulation being analytically embedded in an optimization formulation. The formulation is an integer-linear program, and does not assume fixed cycle lengths or phase sequences. It assumes full information on external inputs, but can be incorporated in a sensor-based environment, as well as in a feedback control framework. The formulation is an integer-linear program that may not be efficiently solved with standard simplex and branch and bound techniques. We discuss network programming formulations to handle the linear platoon dispersion equations and the integer constraints at the intersections. A special-purpose network simplex algorithm for fast solution is also mentioned.

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

Asymptotics of the Christoffel Functions on a Simplex in [formula omitted

We obtain the asymptotics of the Christoffel functions for certain Jacobi-type weight functions on the standard simplex in Rd. We also establish a sharp upper bound of the asymptotics for a large class of weight functions on the simplex.

Interior-point methods for reduced Hessian successive quadratic programming

Typical chemical process optimization problems have a large number of equations, but relatively few degrees of freedom. A reduced Hessian successive quadratic programming (rSQP) algorithm has been shown to be successful in solving these types of models efficiently. While the rSQP algorithm reduces the dimension of the QP subproblem solved at each iteration, the number of inequality constraints could still become quite large. As a result, there is still a potentially large combinatorial expense to solving large NLPs with rSQP, as the solution of the QP subproblem is the bottle-neck of the SQP algorithm. To overcome this combinatorial expense, interior-point (IP) methods have been shown to solve large linear programs much faster than the standard simplex method. These interior-point techniques have also been extended to solve quadratic programming problems. For these reasons, we describe an IP method to solve the QP subproblem for rSQP at each iteration. The primal-dual interior-point method is described as well as some higher-order strategies designed to improve convergence. These higher order strategies are evaluated, and also a comparison of several linear equation solvers is made, since the solution of a linear system is required for each interior-point iteration. Finally, we evaluate and compare the new interior-point QP formulation with two standard active-set solvers, QPKWIK and QPOPT, on some industrial-strength NLP problems.

Network Traffic Signal Optimization Formulation with Embedded Platoon Dispersion Simulation

The existing network traffic signal optimization formulations usually do not include traffic flow models, except for control schemes such as SCOOT that use simulation for heuristic optimization. Other conventional models normally use isolated intersection optimization with traffic arrival prediction using detector information, or optimization schemes based on green bandwidth. A complete formulation of the problem is presented, including explicit constraints to model the movement of traffic along the streets between the intersections in a time-expanded network, as well as constraints to capture the permitted movements from modern signal controllers. The platoon dispersion model used is the well-known Robertson’s model, which forms linear constraints. Thus, it is a rare example of a traffic simulation being analytically embedded in an optimization formulation. The formulation is an integer-linear program and does not assume fixed cycle lengths or phase sequences. It assumes full information on external inputs but can be incorporated in a sensor-based environment as well as in a feedback control framework. The formulation is an integer-linear program that may not be efficiently solved with standard simplex and branch and bound techniques. Network programming formulations to handle the linear platoon dispersion equations and the integer constraints at the intersections are discussed. A special-purpose network simplex algorithm for fast solutions is also mentioned.

Genetic engineering via negative fitness:Evolutionary dynamics for global optimization

In this paper, we provide a novel interpretation of a global optimization procedure forstandard quadratic problems (QPs) in terms of selection models. A standard QP consists ofmaximizing a quadratic form over the standard simplex. Recently, local solutions to standardQPs have been obtained with the help of replicator dynamics, which were designed to modelevolutionary processes. Following trajectories under these dynamics, one obtains a sequenceof feasible points with strictly increasing objective values, which approach stationary points.We present regularity conditions which ensure that the limiting points are indeed localsolutions, so that jamming could be avoided. Genetic engineering via negative fitness is ablock pivoting procedure which either delivers a certificate for global optimality of thecurrent local solution, or else enables an escape from its basin of attraction, if it is inefficient.The name originates from the way the block pivot is obtained: via minimization (rather thanmaximization) of a subproblem. We also elaborate on an important application, the searchfor a maximum clique in an undirected graph, and report both theoretical and empiricalresults.

A warm-start dual simplex solution algorithm for the minimum flow networks with postoptimality analyses

This paper presents an algorithm for finding the minimum flow in general ( s, t) networks with m directed arcs. The minimum flow problem (MFP) arises in many transportation and communication systems. Here, we construct a linear programming (LP) formulation of MFP and develop a simplex-type algorithm to find a minflow. Unlike other simplex-like algorithms, the algorithm developed here starts with an incomplete Basic Variable Set (BVS) initially and then fills-up the BVS completely while pushing toward an optimal vertex. If this results in pushing too far into infeasibility, the next step pulls the solution back to feasibility. Both phases use the Gauss-Jordan pivoting transformation used in the standard simplex and dual simplex algorithms. The proposed approach has some common features with Dantzig's self-dual simplex algorithm. We have avoided, however, the need for extra variables (slack and surplus) for equality constraints, as well as an artificial constraint for the self-dual algorithm for initial phase and the dual simplex, respectively. The proposed Push phase takes at most 2 m − 1 iterations to complete a tree (this augmentation may not be feasible). An infeasible path to the optimal solution contains at most 2 m − 1 iterations; therefore in theory, the algorithm needs a sequence of at most 4 m − 2 iterations. Furthermore, the algorithm developed here makes available the full power of LP perturbation analysis and facilitates introducing network structural changes and side constraints also. It can also detect clerical errors in data entry which may make the problem infeasible or unbounded. It is assumed that the reader is familiar with LP terminology.

Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks.

Polytopes related to the l∞-distance between vectors

Let K=Kn×Kn×[0,1]⊂R2n+1 where Kn is the standard simplex in Rn, i.e., Kn={x∈R+n:∑j=1nxj=1}. We consider the set P⊆R2n+1 given by P={(x,y,z)∈K:‖x−y‖∞⩽z} (where ‖x−y‖∞=maxj ⩽n|xj−yj|). These polytopes are of interest in integer linear programming problems that arise in, e.g. certain statistical estimation problems. We determine all the vertices of P and give a complete linear description of the polytope M being the convex hull of all integer points in P. Relations to a new polyhedral norm is pointed out and some properties of this norm and its unit ball are given.

On Standard Quadratic Optimization Problems

A standard quadratic optimization problem (QP) consists of finding (global) maximizers of a quadratic form over the standard simplex. Standard QPs arise quite naturally in copositivity-based proced...

Study of polymerization of acrylic bone cement: effect of HEMA and EGDMA.

The polymerization reaction of standard surgical Simplex-P radiopaque bone cement was investigated by differential scanning calorimetry to determine the influence of hydroxyethyl methacrylate (HEMA) and ethylene glycol dimethacrylate (EGDMA) on the polymerization reaction. From the kinetic analysis, the polymerization reaction of the modified acrylic bone cement was found to be approximately a first-order reaction. The reaction rate constants (k) were determined. It was found that the effects of HEMA and EGDMA contents on the rate and the heat of polymerization can be explained by the frequency factor and the activation energy. An increase in HEMA content tends to result in an increase in the values of both frequency factor and activation energy, whereas an increase in EGDMA content tends to induce a decrease in the frequency factor and activation energy.

Global Escape Strategies for Maximizing QuadraticForms over a Simplex

Consider the problem of maximizing a quadratic form over the standard simplex. Problems of this type occur, e.g., in the search for the maximum (weighted) clique in an undirected graph. In this paper, copositivity-based escape procedures from inefficient local solutions are rephrased into lower-dimensional subproblems which are again of the same type. As a result, an algorithm is obtained which tries to exploit favourable data constellations in a systematic way, and to avoid the worst-case behaviour of such NP-hard problems whenever possible. First results on finding large cliques in DIMACS benchmark graphs are encouraging.

Evolution towards the Maximum Clique

As is well known, the problem of finding a maximum clique in a graph is NP-hard. Nevertheless, NP-hard problems may have easy instances. This paper proposes a new, global optimization algorithm which tries to exploit favourable data constellations, focussing on the continuous problem formulation: maximize a quadratic form over the standard simplex. Some general connections of the latter problem with dynamic principles of evolutionary game theory are established. As an immediate consequence, one obtains a procedure which consists (a) of an iterative part similar to interior-path methods based on the so-called replicator dynamics; and (b) a routine to escape from inefficient, locally optimal solutions. For the special case of finding a maximum clique in a graph where the quadratic form arises from a regularization of the adjacence matrix, part (b), i.e. escaping from maximal cliques not of maximal size, is accomplished with block pivoting methods based on (large) independent sets, i.e. cliques of the complementary graph. A simulation study is included which indicates that the resulting procedure indeed has some merits.

An exterior-point method for linear programming problems

This paper proves the convergence of an algorithm for solving linear programming problems inO(mn 2) arithmetic operations. The method is called an exterior-point procedure, because it obtains a sequence of approximations falling outside the setU of feasible solutions. Each iteration consists of a single step within some constraining hyperplane, followed by one or more projections which force the new approximation to fall within some envelope aboutU. The paper also discusses several numerical applications. In some types of problems, the method is considerably faster than a standard simplex method program when the size of the problem is sufficiently large.

Finite-dimensional discrete linear stochastic accelerated-time systems and their application to quadratic stochastic dynamical systems

The limiting behavior of the trajectories {x (n) } of linear discrete stochastic systems of the form (K, P an+b ) n∈N , whereK is the standard simplex in ℝ N ,P:ℝ N → ℝ N is a linear operator,PK ⊂K,a ∈ ft,b ∈ ℤ,a+b>0, is described. An application to a class of quadratic stochastic dynamical systems is considered.

Geometrically convergent projection method in matrix games

We propose a method which evaluates the solution of a matrix game. We reduce the problem of the search for the solution to a convex feasibility problem for which we present a method of projection onto an acute cone. The algorithm converges geometrically. At each iteration, we apply a combinatorial algorithm in order to evaluate the projection onto the standard simplex.

A method for localizing EEG sources in realistic head models.

A computationally practical method for performing moving dipole calculations to localize EEG sources in realistic, boundary element (integral equation) type of head models is presented. This method makes use of a rapid method of solving the forward problem of calculating the EEG's produced by a dipole in a realistic head model. This rapid forward calculation method allows the use of standard Simplex search techniques to solve the inverse problem of localizing electrical sources in the brain from EEG's measured on the scalp.

Multicollinearity and Leverage in Mixture Experiments

The impact of leverage, influential observations, and multicollinearity in mixture experiments is discussed. By example, it is shown that consideration of these factors is important in the practical selection of a mixture design. Both standard simplex..

Stochastic Dedication: Designing Fixed Income Portfolios Using Massively Parallel Benders Decomposition

Drawing on recent developments in discrete time fixed income options theory, we propose a stochastic programming procedure, which we call stochastic dedication, for managing asset/liability portfolios with interest rate contingent claims. The model uses scenario generation to combine deterministic dedication techniques with stochastic duration matching methods, and provides the portfolio manager with a risk/return Pareto optimal frontier from which a portfolio may be selected based on individual risk attitudes. We employ a fixed income risk metric that can be interpreted as the fair market value of a collection of interest rate options that eliminates bankruptcy risk from the asset/liability portfolio. We incorporate this metric into a risk/return stochastic optimization model, using a binomial lattice sampling procedure to construct interest rate paths and cash flow streams from an arbitrage-free term structure model. The resulting parametric linear program has a particularly simple subproblem structure, and we have been able to solve it using resource-directed decomposition on a massively parallel computer system, the Connection Machine CM-2. We take a novel approach that uses a standard serial simplex method to solve the master problem, but generates scenarios and Benders cuts in a massively parallel manner. We discuss the performance of this implementation and present the results for a simple pension fund immunization problem.

Isospectral flows and linear programming

AbstractBrockett has studied the isospectral flow Ḣ = [H, [H, N]], with [A, B] = AB ∔ BA, on spaces of real symmetric matrices. The flow diagonalises real symmetric matrices and can be used to solve linear programming problems with compact convex constraints. We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an ε-neighbourhood of the optimum. An interior point algorithm for the standard simplex is analysed in detail and a comparison is made with a continuous time version of Karmarkar algorithm.