Quadratic Function Subject Research Articles

Inverse kinematics of asymmetric octahedral variable geometry truss manipulator with obstacle avoidance through inexact interior point optimization

This article studies the inverse kinematics for asymmetric octahedral variable geometry truss manipulator with obstacle avoidance. The inverse kinematics problem is cast as a nonconvex optimization that having quadratic objective function subject to quadratic constraints. This article uses an inexact interior point optimization to solve it, which is developed on the basis of the imprecise algorithm Ipopt. According to the particularity of our actual optimization problem, each iteration undergoes specific modifications so as to minimize the memory consumption as well as computation time. Utilizing the sparse and binary characteristics of the coefficient matrix, respectively, the algorithm allocates the computation to the finite sparse matrix vector multiplication and changes the storage form, which greatly reduces the memory space. Based on the unique rules of inverse kinematics, the iteration direction of the algorithm becomes more clear. With the aid of mechanical constraints inherent in the manipulator, the algorithm omits the feasibility recovery part that embedded in the solver Ipopt. All these make us save the operation time greatly while utilizing Ipopt algorithm. To demonstrate the effectiveness of the proposed approach, the scheme was applied to obstacle avoidance inverse kinematics of variable geometry truss manipulator with three modules.

An Algorithm for Solving Indefinite Quadratic Programming Problems

In this paper, we give in section (1) compact description of the algorithm for solving general quadratic programming problems (that is, obtaining a local minimum of a quadratic function subject to inequality constraints) is presented. In section (2), we give practical application of the algorithm, we also discuss the computation work and performing by the algorithm and try to achieve efficiency and stability as possible as we can. In section (3), we show how to update the QR-factors of , when the tableau is complementary ,we give updating to the LDLT-Factors of . In section (4) we are not going to describe a fully detailed method of obtaining an initial feasible point, since linear programming literature is full of such techniques.

Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint

In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that under a mild assumption, this problem can be solved by solving a linearly constrained convex univariate minimization problem. Finally, the superior efficiency of the new approach compared to the known semidefinite relaxation and a known approach from the literature is demonstrated by solving several randomly generated test problems.

Stochastic Optimal Control of Finite Ensembles of Nanomagnets

We control ferromagnetic N-spin dynamics in the presence of thermal fluctuations by minimizing a quadratic functional subject to the stochastic Landau–Lifshitz–Gilbert equation. Existence of a weak solution of the stochastic optimal control problem is shown. The related first order optimality conditions consist of a coupled forward–backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the new stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise.

An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables

This paper proposes and analyzes an affine scaling trust-region method with line search filter technique for solving nonlinear optimization problems subject to bounds on variables. At the current iteration, the trial step is generated by the general trust-region subproblem which is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint. Both trust-region strategy and line search filter technique will switch to trail backtracking step which is strictly feasible. Meanwhile, the proposed method does not depend on any external restoration procedure used in line search filter technique. A new backtracking relevance condition is given which is weaker than the switching condition to obtain the global convergence of the algorithm. The global convergence and fast local convergence rate of this algorithm are established under reasonable assumptions. Preliminary numerical results are reported indicating the practical viability and show the effectiveness of the proposed algorithm.

SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and Hertog (Math. Program. 143(1-2):1-29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. Compared with the state-of-the-art approach to reformulate the GTRS as a semi-definite programming problem, our SOCP reformulation delivers a much faster solution algorithm. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. Our methods also enable us to obtain simplified versions of the classical S-lemma, the S-lemma with equality, and the S-lemma with interval bounds.

A Proposed Method to Solve Quadratic Fractional Programming Problem by Converting to Double Linear Programming

Quadratic fractional program is an optimization problem which solving the problem by minimizes or maximizes a quadratic fractional objective function subject to finite number of linear inequality (equality constraints) . In our paper, we proposed a new method to solve quadratic fractional programming problem (QFPP),the objective function of quadratic optimization has linear factorized as product of two linear functions, the two positive linear functions solved separately by using simplex method. These are useful in solving the problem in multi-application, like economics, hospital and health, engineering problem, financial planning etc.In our paper, it was addressed to a variety of examples and the results were encouraging and accurate comparison with other methods with ease in the solution.

A multi-start iterated local search algorithm for the generalized quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) is a variant of the classical knapsack problem where multiple knapsacks are considered and the objective is to maximize a quadratic objective function subject to capacity constraints. The generalized quadratic multiple knapsack problem (G-QMKP) extends the QMKP by considering the setups, assignment conditions and the knapsack preferences of the items. In this study, a multi-start iterated local search algorithm (MS-ILS) in w the variable neighborhood descent (VND) algorithm is integrated with an adaptive perturbation mechanism is proposed to solve the G-QMKP. The multi-start implementation together with the adaptive perturbation mechanism enables the search process to explore different search regions in the search space while VND is applied to obtain high-quality solutions from the examined regions. Another remarkable feature of MS-ILS is its simplicity and adaptiveness that ease its implementation. The proposed MS-ILS is tested on G-QMKP benchmark instances derived from the literature. The results indicate that the developed MS-ILS can produce high-quality solutions for the G-QMKP. Particularly, it has been observed that the developed MS-ILS has improved the best known solutions for 35 out of 48 large-size G-QMKP instances.

A dual gradient-projection method for large-scale strictly convex quadratic problems

The details of a solver for minimizing a strictly convex quadratic objective function subject to general linear constraints are presented. The method uses a gradient projection algorithm enhanced with subspace acceleration to solve the bound-constrained dual optimization problem. Such gradient projection methods are well-known, but are typically employed to solve the primal problem when only simple bound-constraints are present. The main contributions of this work are threefold. First, we address the challenges associated with solving the dual problem, which is usually a convex problem even when the primal problem is strictly convex. In particular, for the dual problem, one must efficiently compute directions of infinite descent when they exist, which is precisely when the primal formulation is infeasible. Second, we show how the linear algebra may be arranged to take computational advantage of sparsity that is often present in the second-derivative matrix, mostly by showing how sparse updates may be performed for algorithmic quantities. We consider the case that the second-derivative matrix is explicitly available and sparse, and the case when it is available implicitly via a limited memory BFGS representation. Third, we present the details of our Fortran 2003 software package DQP, which is part of the GALAHAD suite of optimization routines. Numerical tests are performed on quadratic programming problems from the combined CUTEst and Maros and Meszaros test sets.

Inverse kinematics for the variable geometry truss manipulator via a Lagrangian dual method

This article studies the inverse kinematics problem of the variable geometry truss manipulator. The problem is cast as an optimization process which can be divided into two steps. Firstly, according to the information about the location of the end effector and fixed base, an optimal center curve and the corresponding distribution of the intermediate platforms along this center line are generated. This procedure is implemented by solving a non-convex optimization problem that has a quadratic objective function subject to quadratic constraints. Then, in accordance with the distribution of the intermediate platforms along the optimal center curve, all lengths of the actuators are calculated via the inverse kinematics of each variable geometry truss module. Hence, the approach that we present is an optimization procedure that attempts to generate the optimal intermediate platform distribution along the optimal central curve, while the performance index and kinematic constraints are satisfied. By using the Lagrangian duality theory, a closed-form optimal solution of the original optimization is given. The numerical simulation substantiates the effectiveness of the introduced approach.

Local nonglobal minima for solving large-scale extended trust-region subproblems

We study large-scale extended trust-region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust-region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving large-scale TRS problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS by solving at most three generalized eigenvalue problems.

An efficient algorithm for solving the generalized trust region subproblem

In this paper, we consider the interval bounded generalized trust region subproblem (GTRS) which is the problem of minimizing a general quadratic function subject to an upper and lower bounded general quadratic constraint. Under the assumption that two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence, a diagonalization-based algorithm is introduced to solve it by showing that GTRS is indeed equivalent to a linearly constrained convex univariate problem. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with the extended Rendl–Wolkowicz algorithm due to Pong and Wolkowicz.

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

This article considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP has been developed to solve this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is computed using the variable fixing method or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the dual function and computing the solution of the dual problem. Also, a new version of the variable fixing algorithm is developed that is convergent even when the objective Hessian is not strictly positive definite. The hybrid algorithm NAPHEAP that uses a Newton-type method (variable fixing method, secant method, or Newton's method) to bracket a root, followed by a heap-based monotone break point search, can be faster than a Newton-type method by itself, as demonstrated in the numerical experiments.

Interval Type-2 Fuzzy Model Predictive Control of Nonlinear Networked Control Systems

In this paper, the problem of fuzzy predictive control of nonlinear networked control systems subject to parameter uncertainties and defective communication links is studied. Stochastic variables with Bernoulli random binary distribution are used to represent the defective communication links with packets loss occurring intermittently between the controller and the physical plant. An interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model is employed to describe the nonlinear plant subject to parameter uncertainties, which can be captured with the lower and upper membership functions. The IT2 fuzzy model and IT2 fuzzy controller are not required to share the same lower and upper membership functions. In order to design the state-feedback fuzzy model predictive controller, an optimization problem which minimizes the upper bound of a quadratic objective function subject to input constraints and packets dropout is formulated and solved at every sampling instant in the finite time horizon. By introducing some slack matrices, less conservative conditions are developed for system stability analysis. Two examples are given to demonstrate the effectiveness and merits of the proposed new design techniques.

Minimizing quadratic functions with semidefinite Hessian subject to bound constraints

The MPRGP (modified proportioning with reduced gradient projections) algorithm for minimization of the strictly convex quadratic function subject to bound constraints is adapted to the solution of problems with a semidefinite Hessian A. The adapted algorithm accepts the decrease directions that belong to the null space of A and generates the iterates that are proved to minimize the cost function. The paper examines specific features of the solution of the problems with convex, but not necessarily strictly convex Hessian. The performance of the algorithm is demonstrated by the solution of a semi-coercive contact problem of elasticity and a 3D particle dynamics problem. The results are compared with those obtained by the spectral projected gradient method and the projected-Jacobi method.

PAPMSC: Power-Aware Performance Management Approach for Virtualized Web Servers via Stochastic Control

As green computing is becoming a popular computing paradigm, the performance of energy-efficient data center becomes increasingly important. This paper proposes power-aware performance management via stochastic control method (PAPMSC), a novel stochastic control approach for virtualized web servers. It addresses the instability and inefficiency issues due to dynamic web workloads. It features a coordinated control architecture that optimizes the resource allocation and minimizes the overall power consumption while guaranteeing the service level agreements (SLAs). More specifically, due to the interference effect among the co-located virtualized web servers and time-varying workloads, the relationship between the hardware resource assignment to different virtual servers and the web applications' performance is considered as a coupled Multi-Input-Multi-Output (MIMO) system and formulated as a robust optimization problem. We propose a constrained stochastic linear-quadratic controller (cSLQC) to solve the problem by minimizing the quadratic cost function subject to constraints on resource allocation and applications' performance. Furthermore, a proportional controller is integrated to enhance system stability. In the second layer, we dynamically manipulate the physical frequency for power efficiency using an adaptive linear quadratic regulator (ALQR). Experiments on our testbed server with a variety of workload patterns demonstrate that the proposed control solution significantly outperforms existing solutions in terms of effectiveness and robustness.

Computational method for the real-time calculation of the full-body muscle load distribution

A method is described for minimising a quadratic function subject to equality and inequality constraints. This approach is applicable to solving the full-body muscle load distribution problem and calculating joint contact loads. It has been found that this approach can provide the solution on modest computing facilities and in significantly less time than using active set and interior point quadratic programming techniques. Hence the approach is suitable for providing real-time feedback to subjects undergoing biomechanical analysis of muscle, skeletal and joint loadings.

Local Minima of Quadratic Functionals and Control of Hydro-electric Power Stations

We consider a control problem for a cascade of hydro-electric power stations, where some of the stations have reversible turbines. Our aim was to optimize the profit of power production satisfying restrictions on the water level in the reservoirs. From the mathematical point of view, this consists in minimizing an infinite-dimensional quadratic functional subject to cone constraints. Sufficient conditions of optimality for the abstract problem are derived and are then specialized for our problem. Noteworthy, the restrictions imposed on the power stations problem are in the form of control constraints and pure state constraints. The particular case of one power station is analyzed in detail, showing that reversible turbines always improve the profit.

An efficient hybrid heuristic method for the 0-1 exact k-item quadratic knapsack problem

The 0-1 exact k-item quadratic knapsack problem (E - kQKP) consists of maximizing a quadratic function subject to two linear constraints: the first one is the classical linear capacity constraint; the second one is an equality cardinality constraint on the number of items in the knapsack. Most instances of this NP-hard problem with more than forty variables cannot be solved within one hour by a commercial software such as CPLEX 12.1. We propose therefore a fast and efficient heuristic method which produces both good lower and upper bounds on the value of the problem in reasonable time. Specifically, it integrates a primal heuristic and a semidefinite programming reduction phase within a surrogate dual heuristic. A large computational experiments over randomly generated instances with up to 200 variables validates the relevance of the bounds produced by our hybrid dual heuristic, which yields known optima (and prove optimality) in 90% (resp. 76%) within 100 seconds on the average.

Quasi-min-max dynamic output feedback MPC for LPV systems

This paper presents a dynamic output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on quasi–min–max scheme. At each time instant, an optimal control sequence is computed by minimising the upper bound of an infinite horizon quadratic objective function subject to input constraints using linear matrix inequalities (LMIs). A sufficient condition is derived to guarantee robust stability of the closed–loop system and an iterative algorithm involving convex optimisations is presented to solve the online optimisation problem. Finally, two numerical examples were used to demonstrate the effectiveness of the proposed results.