Quadratic Search Research Articles

Performance analysis of variant working vacation queue with balking and reneging

This paper studies the effect of balking and reneging in a single server queue under variant working vacation policy. Arriving customers balk (do not join the queue) with a probability and renege (leave the queue after joining) according to exponential distribution. Under the variant vacation policy the server leaves for a working vacation as soon as the system becomes empty. The server takes at most J working vacations until he finds at least one customer in the queue on return from a working vacation. If customers are not found in the queue by the end of Jth vacation, the server remains idle in the system. The system length probabilities are derived using matrix form solution. Some performance measures of the model and cost analysis through quadratic fit search method have been discussed. Finally, the sensitivity analysis is carried out through some numerical experiments.

Hybrid Approach to Optimize the Cluster Flying Orbit for Fractionated Spacecraft Based on PSO-SQP Algorithm

This template This paper investigates the optimal cluster flight orbit design issue for the fractionated spacecraft according to three main goals: keeping the cluster as stable as possible, preventing the collisions within the cluster, maintaining the inter-satellite distance within the maximum region. Firstly, the relative orbital elements are adopted to describe the relative motion. Then, the formation design requirements are formulated in terms of the relative orbital elements, the constrained optimization problem is solved using the hybrid particle swarm optimization algorithm integrated with sequential quadratic programming local search. The simulation results show that the hybrid PSO-SQP algorithm is effective.

A novel PID controller tuning method based on optimization technique

An approach for parameter estimation of proportional-integral-derivative (PID) control system using a new nonlinear programming (NLP) algorithm was proposed. SQP/IIPM algorithm is a sequential quadratic programming (SQP) based algorithm that derives its search directions by solving quadratic programming (QP) subproblems via an infeasible interior point method (IIPM) and evaluates step length adaptively via a simple line search and/or a quadratic search algorithm depending on the termination of the IIPM solver. The task of tuning PI/PID parameters for the first- and second-order systems was modeled as constrained NLP problem. SQP/IIPM algorithm was applied to determining the optimum parameters for the PI/PID control systems. To assess the performance of the proposed method, a Matlab simulation of PID controller tuning was conducted to compare the proposed SQP/IIPM algorithm with the gain and phase margin (GPM) method and Ziegler-Nichols (ZN) method. The results reveal that, for both step and impulse response tests, the PI/PID controller using SQP/IIPM optimization algorithm consistently reduce rise time, settling-time and remarkably lower overshoot compared to GPM and ZN methods, and the proposed method improves the robustness and effectiveness of numerical optimization of PID control systems.

Matching Heterogeneous Agents with a Linear Search Technology

Steady state equilibria in heterogeneous agent matching models with search frictions have been shown to exist in Shimer and Smith (2000) under the assumption of a quadratic search technology. We extend their analysis to the com- monly investigated linear search technology.

Genetic/quadratic search algorithm for plant economic optimizations using a process simulator

The genetic/quadratic search algorithm (GQSA) is a hybrid genetic algorithms (GA) for optimizing plant economics when a process simulator models the plant. By coupling a regular GA with an algorithm based upon a quadratic search, the required number of objective function evaluations for obtaining an acceptable solution decreases significantly in most cases. The GQSA combines advantages of GA and quadratic search techniques, e.g. determining a global optimum for a problem with a high probability for discontinuous as well as non-convex optimization problems while at the same time providing faster convergence than conventional GA. The performance of both the GQSA and the GA was compared using four different test functions and an economic optimization problem for a turbo-expander process. Numerical test results indicate that the convergence of the GQSA is either better than or at least comparable to those of GA for all tests employing the same genetic parameters.

Technique for incorporating the density functional Hessian into the geometry optimization of biomolecules, solvated molecules, and large floppy molecules.

Traditional geometry optimization methods require the gradient of the potential surface, together with a Hessian which is often approximated. Approximation of the Hessian causes difficulties for large, floppy molecules, increasing the number of steps required to reach the minimum. In this article, the costly evaluation of the exact Hessian is avoided by expanding the density functional to second order in both the nuclear and electronic variables, and then searching for the minimum of the quadratic functional. The quadratic search involves the simultaneous determination of both the geometry step and the associated change in the electron density matrix. Trial calculations on Taxol indicate that the cost of the quadratic search is comparable to the cost of the density functional energy plus gradient. While this procedure circumvents the bottleneck coupled-perturbed step in the evaluation of the full Hessian, the second derivatives of the electron-repulsion integrals are still required for atomic-orbital-based calculations, and they are presently more expensive than the energy plus gradient. Hence, we anticipate that the quadratic optimizer will initially find application in fields in which existing optimizers breakdown or are inefficient, particularly biochemistry and solvation chemistry.

Methods for optimizing large molecules. II. Quadratic search

Geometry optimization has become an essential part of quantum-chemical computations, largely because of the availability of analytic first derivatives. Quasi-Newton algorithms use the gradient to update the second derivative matrix (Hessian) and frequently employ corrections to the quadratic approximation such as rational function optimization (RFO) or the trust radius model (TRM). These corrections are typically carried out via diagonalization of the Hessian, which requires O(N3) operations for N variables. Thus, they can be substantial bottlenecks in the optimization of large molecules with semiempirical, mixed quantum mechanical/molecular mechanical (QM/MM) or linearly scaling electronic structure methods. Our O(N2) approach for solving the equations for coordinate transformations in optimizations has been extended to evaluate the RFO and TRM steps efficiently in redundant internal coordinates. The regular RFO model has also been modified so that it has the correct size dependence as the molecular systems become larger. Finally, an improved Hessian update for minimizations has been constructed by combining the Broyden–Fletcher–Goldfarb–Shanno (BFGS) and (symmetric rank one) SR1 updates. Together these modifications and new methods form an optimization algorithm for large molecules that scales as O(N2) and performs similar to or better than the traditional optimization strategies used in quantum chemistry.

Robust Recursive Quadratic Programming Algorithm Model with Global and Superlinear Convergence Properties

A new, robust recursive quadratic programming algorithm model based on a continuously differentiable merit function is introduced. The algorithm is globally and superlinearly convergent, uses automatic rules for choosing the penalty parameter, and can efficiently cope with the possible inconsistency of the quadratic search subproblem. The properties of the algorithm are studied under weak a priori assumptions; in particular, the superlinear convergence rate is established without requiring strict complementarity. The behavior of the algorithm is also investigated in the case where not all of the assumptions are met. The focus of the paper is on theoretical issues; nevertheless, the analysis carried out and the solutions proposed pave the way to new and more robust RQP codes than those presently available.

Modifying the BFGS update by a new column scaling technique

LetB be a positive definite symmetric approximation to the second derivative matrix of the objective function of a minimization calculation. We study modifications of the BFGS method that apply a scaling technique to the columns of a conjugate direction matrixZ satisfyingZ T BZ = I. For a simple scaling technique similar to the ones considered by Powell (1987) and (1989) we show that, due to a two iteration cycle, linear convergence can occur when the method is applied to a quadratic function and Wolfe's line search is employed, although for exact line searches quadratic termination can be proved. We then suggest a different scaling technique that prevents certain columns thought to contain important curvature information from being scaled. For this algorithm we prove global and superlinear convergence and demonstrate the superiority of our method over the BFGS formula on a range of illconditioned optimization problems. Moreover, we present an implementation of our algorithm that requires only 3n 2 +O(n) multiplications per iteration.

An efficient interfacing of golden-section and quadratic searches

A simultaneous improvement in robustness and efficiency of a univariate optimization can be achieved through a combination of golden-section and quadratic interpolation searches. Using a least-squares approximation during a modified golden-section search, it is possible to determine when the switch-over of search strategies is best accomplished. On completion of the golden-section search, the minimizing point is estimated with improved accuracy from the least squares which can be used as a good starting point for the following quadratic search. Explicit formulas are given resulting in minimal programming and computational overheads. Attention is paid to implementing the algorithm on simple processors. An easily computable bound on error estimate of the function minimum is derived.

An improved hash code for scatter storage

Introduced is a hash coding method based on fixed-point division rather than multiplication or logical operations. This new method allows the hash table to have almost any length. Also a new method of handling collisions is discussed. Known as quadratic search, this method is faster than random search and free from the “clusters” that build up with a linear search.

The Single Source Weber Problem — Survey and Extensions

Alfred Weber reduced the problem of locating a firm under the conditions that all cost elements other than those which vary regularly with space are held constant, to that of determining the point(s) of minimum transportation costs. This paper presents a survey of the techniques used to solve this problem and gives an efficient Quadratic Approximation Search Technique which was coded in FORTRAN IV and tested on a number of problems for it.

Quadratic search for hash tables of sizes P n

It has previously been claimed [1 and 2] that the quadratic hash table search method of Maurer cannot usefully be applied to tables of size 2 n . This is not so; the method can in fact be applied to tables of size p n for any prime p . It is shown below that rather simple conditions on the coefficients suffice to guarantee that all table locations will be examined once and only once. Specifically, if the equation is k + bi 2 + ai mod p n (*) where k is the initial hash address and 0 ≤ i < p n , then, if p divides b but not a , the range of values is all the least positive residues of p n . To prove that all values are covered, we consider some fixed value, say k + bi 2 0 + ai 0 mod p n and ask, what conditions must be true if the congruence equation k + bi 2 + ai ≡ k + bi 0 2 + ai 0 mod p n is to have solutions i , 0 ≤ i < p n , other than i 0 ?

A full table quadratic search method eliminating secondary clustering

A quadratic search method for hash tables whose size is a power of 2 is presented. It allows full table searching, and yet eliminates secondary clustering, which occurs when different keys hashed initially to the same location proceed to trace through the same sequence of locations. Comparisons with other methods are made experimentally.

Quadratically convergent algorithms and one-dimensional search schemes

In this paper, the performances of three quadratically convergent algorithms coupled with four one-dimensional search schemes are studied through several nonquadratic examples. The algorithms are the rank-one algorithm (Algorithm I), the projection algorithm (Algorithm II), and the Fletcher-Reeves algorithm (Algorithm III). The search schemes are the exact quadratic search (EQS), the exact cubic search (ECS), the approximate quadratic search (AQS), and the approximate cubic search (ACS). The performances are analyzed in terms of number of iterations and number of equivalent function evaluations for convergence. From the numerical experiments, the following conclusions are found: (a) while the number of iterations generally increases by relaxing the search stopping condition, the number of equivalent function evaluations decreases; therefore, approximate searches should be preferred to exact searches; (b) the numbers of iterations for ACS, ECS, and EQS are about the same; therefore, the use of more sophisticated, higher order search schemes is not called for; the present ACS scheme, modified so that only the function, instead of the gradient, is used in bracketing the minimal point, could prove to be most desirable in terms of the number of equivalent function evaluations; (c) for Algorithm I, ACS and AQS yield almost identical results; it is believed that further improvements in efficiency are possible if one uses a fixed stepsize approach, thus bypassing the one-dimensional search completely; (d) the combination of Algorithm II and ACS exhibits high efficiency in treating functions whose order is higher than two and whose Hessian at the minimal point is singular; and (f) Algorithm III, even with the best search scheme, is inefficient in treating functions with flat bottoms; it is doubtful that the simplicity of its update will compensate for its inefficiency in such pathological cases.

Full table quadratic searching for scatter storage

The quadratic residue search method for hash tables avoids much of the clustering experienced with a linear search method. The simple quadratic search only accesses half the table. It has been shown that when the length of the table is a prime of the form 4 n + 3, where n is an integer, the whole table may be accessed by two quadratic searches plus a separate access for the original entry point. A search method is presented which is computationally simple, has all the advantages of the quadratic search, and yet accesses all the table in one sweep.