An interior-point QP algorithm for structural optimization

In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O(nlog((x0)Ts0/e)) which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.

In this paper, we present a long-step infeasible primal-dual path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. In contrast to the authors’ previous paper [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1) (2006), pp. 287–310], we propose a new linear system, which we refer to as the hybrid augmented normal equation (HANE), to determine the primal-dual search directions. Since the iterative linear solver can only generate an approximate solution to the HANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a recipe to compute an inexact primal-dual search direction, based on a suitable approximate solution to the HANE. The second difference between this paper and [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1)(2006), pp. 287–310] is that, instead of using the maximum weight basis (MWB) preconditioner in the aforesaid recipe for constructing the inexact search direction, this paper proposes the use of any member of a whole class of preconditioners, of which the MWB preconditioner is just a special case. The proposed recipe allows us to: (i) establish a polynomial bound on the number of iterations performed by our path-following algorithm and (ii) establish a uniform bound, depending on the quality of the preconditioner, on the number of iterations performed by the iterative solver.

A polynomial arc-search interior-point algorithm for convex quadratic programming

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.

Although the classical LQR design method has been very successful in real world engineering designs, in some cases, the classical design method needs modifications because of the saturation in actuators. This modified problem is sometimes called the constrained LQR design. For discrete systems, the constrained LQR design problem is equivalent to a convex quadratic programming problem with box constraints. We will show that the interior-point method is very efficient for this problem because an initial interior point is available, a condition which is not true for general convex quadratic programming problem. We will devise an effective and efficient algorithm for the constrained LQR design problem using the special structure of the box constraints and a recently introduced arc-search technique for the interior-point algorithm. We will prove that the algorithm is polynomial and has the best-known complexity bound for the convex quadratic programming. The proposed algorithm is implemented in MATLAB. An example for the constrained LQR design is provided to show the effectiveness and efficiency of the design method. The proposed algorithm can easily be used for model predictive control.

In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighborhoods: a small one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields a (practical) long-step algorithm. Both algorithms allow for the Newton equation system to be solved inexactly. For both algorithms we will provide conditions for the level of error acceptable in the Newton equation and establish the worst-case complexity results.

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.

We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameterα (so that we must have 0 <α ≤ 1) the best possible guaranteeδn(α) inn dimensional space is essentially ln 2 ≃ 0.69; and to achieve this we must takeα about 1. Indeed we show the precise result thatδn(α) equals ln(1 +α)-ln(1 −α/(n − 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to aboutδ* ≃ 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.

One of the downsides of the massive multiple-input-multiple-output (M-MIMO) system is its computational complexity. Considering that techniques and different algorithms proposed in the literature applied to conventional MIMO may not be well suited or readily applicable to M-MIMO systems, in this paper, the application of different formulations inside the convex optimization framework is investigated. This paper is divided into two parts. In the first part, linear programming, quadratic programming (QP), and semidefinite programming are explored in an M-MIMO environment with high-order modulation and under realistic channel conditions, i.e., considering spatial correlation, error in the channel estimation, as well as different system loading. The bit error rate is evaluated numerically through Monte Carlo simulations. In the second part, algorithms to solve the QP formulation are explored, and computational complexity in terms of floating-point operations (flops) is compared with linear detectors. Those algorithms have interesting aspects when applied to our specific problem (M-MIMO detection formulated as QP), such as the exploitation of the structure of the problem ( simple constraints ) and the improvement of the rate of convergence due to the well-conditioned Gram matrix (channel hardening). The number of iterations is higher when the number of users $K$ becomes similar to the number of base station antennas $M$ (i.e., $K\approx M$ ) than the case $K\ll M$ ; the number of iterations increases slowly as the number of users $K$ and base station antennas $M$ increases while keeping a low system loading. The QP with projected algorithms presented better performance than minimum mean square error detector when $K\approx M$ and promising computational complexity for scenarios with increasing $K$ and low system loading.

There are some characteristics in the surrogate dual for mathematical programming, such as the quasi concavity of the objective function of the dual problem, the surrogate multipliers in the simplex, i.e. \( \sum\nolimits_{i = 1}^m {\lambda _i } = 1 \) , which give much convenient conditions for constructing algorithms for solving the surrogate dual problem. In the theory of surrogate dual, the objective function is expressed only in implicit form, which may bring difficulties to the numerical treatment. In this paper a novel approach on the explicit form of surrogate dual problem, quasi active set and interior point algorithm is presented to solve quadratic programming problem, and some quadratic programming problems are constructed to verify the approach.

Quadratic programming (QP) is a family of methods, techniques, and algorithms that can be used to minimize quadratic objective functions subject to linear constraints. QP shares many combinatorial features with linear programming (LP) and it is often used as the basis of constrained nonlinear programming. An important branch of QP is convex QP where the objective function is a convex quadratic function.

We show how to speed up Karmarkar's linear programming algorithm for the case of multicommodity flows. The special structure of the constraint matrix is exploited to obtain an algorithm for the multicommodity flow problem which requires O(s 3.5 v 2.5 eL) arithmetic operations, each operation being performed to a precision of O (L) bits. Herev is the number of vertices ande is the number of edges in the given network,s is the number of commodities, andL is bounded by the number of bits in the input. We obtain a speed up of the order of (e 0.5/v 0.5)+(e 2.5/v 2.5s2) over Karmarkar's modified algorithm which is substantial for dense networks. The techniques in the paper can also be used to speed up any interior point algorithm for any linear programming problem whose constraint matrix is structurally similar to the one in the multicommodity flow problem.

In this paper, we show that Ye‐Todd‐Mizuno's O(\(\sqrt {nL} \)) ‐iteration homogeneous andself‐dual linear programming (LP) algorithm possesses quadratic convergence of the duality gap to zero. In the case of infeasibility, this shows that a homogenizing variable quadratically converges to zero (which proves that at least one of the primal and dual LP problems is infeasible) and implies that the iterates of the (original) LP variable quadratically diverge. Thus, we have established a complete asymptotic convergence result for interior‐point algorithms without any assumption on the LP problem.

When applying the sequential quadratic programming (SQP) algorithm to topology optimization, using the quasi-Newton methods or calculating the Hessian matrix directly will result in a considerable amount of calculation, making it computationally infeasible when the number of optimization variables is large. To solve the above problems, this paper creatively proposes a method for calculating the approximate Hessian matrix for structural topology optimization with minimum compliance problems. Then, the second-order Taylor expansion transforms the original problem into a series of separable and easy-to-solve convex quadratic programming (QP) subproblems. Finally, the quadratic programming optimality criteria (QPOC) method and the QP solver of MATLAB are used to solve the subproblems. Compared with other sequential quadratic programming methods, the advantage of the proposed method is that the Hessian matrix is diagonally positive definite and its calculation is simple. Numerical experiments on an MBB beam and cantilever beam verify the feasibility and efficiency of the proposed method.

The main viewpoint of land use planning is to resolve the conflict between competing land uses in one place that is carried out based on their economic, social and environmental suitability. Many different approaches have been designed to resolve this conflict but any of them have done. This study aimed to assess accuracy and validity compared four optimization algorithms, including linear programming (LP), simulated annealing (SA), multi-objective land use allocation (MOLA) and multidimensional choice (MDCHOICE) for land use planning in Gorgan Township. At first, land use allocation was done through the LP algorithm. Then, using achieved pixels quantity from LP optimization, the land uses was optimized in term of spatial suitability by other algorithms. The allocation maps generated through these algorithms were compared using statistical methods and landscape metrics. The map provided through MDCHOICE was used as a reference map and that generated through MOLA, SA and LP as a comparative one in a validation process. The similarities and agreements of the two maps were evaluated with various algorithms in IDRISI and MCK software. The results demonstrate that: (1) It follows that the LP algorithm to determine the optimal number of pixels to lead MDCHOICE has helped in the spatial allocation properly. (2) MOLA had a better performance for development in terms of all landscape metrics and SA algorithm was better in forestry land use allocation. Therefore the quality of spatial allocation both of them needs to be improved. (3) Although the LP algorithm is not superior to the algorithms based on the dimensions studied in this research, in terms of achieving the defined goals and the degree of deviation from the number of cells defined for each land use has the least violation. (4) The superiority of MDCHOICE over MOLA, SA and LP and applicability of various kappa statistics for comparing the maps for a more in depth analysis of agreements and disagreements.