Smooth Newton method to absolute value equation based on lower uniform smoothing approximation function

In this paper, an upper uniform smooth approximation function of absolute value function is proposed, and some properties of uniform smooth approximation function are studied. Then, absolute value equation (AVE), Ax - |x| = b, where A is a square matrix whose singular values exceed one, is transformed into smooth optimization problem by using the upper uniform smooth approximation function, and solved by quasi-Newton method. Numerical results in solving given AVE problems demonstrated that our algorithm is valid and superior to lower uniform smooth approximation function.

In this article, we take the piecewise linear equation system \(x-W|x|=b\), which is also known by absolute value equation, where \(W\in {\mathbb R}^ {n\times n}\), \(b\in {\mathbb R}^{n}\) are given and to undetermined the value of \(x\in {\mathbb R}^{n}\). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. (small corrections operated in the pdf file on January 7, 2023)

We propose a hybrid algorithm for solving the NP-hard absolute value equation (AVE): \(Ax-|x|=b\), where \(A\) is an \(n\times n\) square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving iteratively a linear system of equations followed by a linear program. The algorithm was tested on 100 consecutively generated random solvable instances of the AVE with \(n=\) 50, 100, 200, 500 and 1000. The algorithm solved \(100\,\%\) of the test problems to an accuracy of \(10^{-8}\) by solving an average of 2.77 systems of linear equations and linear programs per AVE.

The system of absolute value equation, denoted by AVE, is a non-differentiable NP-hard problem. Many approaches have been proposed during the past decade and most of them focus on reformulating it as complementarity problem and then solve it accordingly. Another approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations. In this paper, we follow this path. In particular, we rewrite it as a system of smooth equations and propose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations. The main contribution of this paper focuses on numerical comparisons which suggest a better choice of smoothing function along with the smoothing-type algorithm.

Non-interior-point smoothing Newton method (SNM) for optimization have been widely studied for over three decades. SNM is a popular approach for solving small- and medium-scale complementarity problem (CP) and many optimization problems. The main purpose of this paper is to revisit the SNM and show that the Hessian matrix in SNM becomes increasingly ill-conditioned while smoothing parameter approaches to zero, which leads to their practical use remains limited due to computational difficulties in solving large-scale CP. To tackle this, we redesign a new smoothing method, called accelerated preconditioned smoothing method (APSM) for the efficient solution of regularized support vector machines in machine learning. With the help of suitable preconditioner, we can correct the ill-conditioning of associated smoothing Hessian matrix and thereby the associated smoothing Hessian equation can be solved in a few of iterations by using iterative methods in linear algebra. Two accelerated techniques are designed in the paper to reduce our computation time. Finally we present numerical experiments to support our theoretical guarantees and test accelerated convergence obtained by APSM. The result showed that APSM computes faster than state of art algorithm without reducing classification accuracy.

We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games. We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria. We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nash equilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasi-concave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium. Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constant-sum games (rank-0 games). It is though incomparable to the subclass of games with fixed rank [16]: Even rank-1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nash equilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.

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.

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.

"Game Theory-Mathematical Models of Conflict." Journal of the Operational Research Society, 31(11), pp. 1050–1051

The problem of regulating air traffic in the en route airspace of the National Airspace System is studied using a Eulerian network model to describe air traffic flow. The evolution of traffic on each edge of the network is modeled by a modified Lighthill-Whitham-Richards partial differential equation. The equation is transformed with a variable change, which makes it linear and enables us to use linear finite difference schemes to discretize the problem. We pose the problem of optimal traffic flow regulation as a continuous optimization program in which the partial differential equation appears in the constraints. We propose a discrete formulation of this problem, which makes all constraints (the discretized partial differential equations, boundary, and initial conditions) linear. Corresponding linear programming and quadratic programming based solutions to this convex optimization program yield globally optimal solutions to various air traffic management objectives. The proposed method is applied to the maximization of aircraft arrivals and minimization of delays in the arrival airspace due to exogenous capacity reductions. The corresponding linear and quadratic programs are solved numerically using CPLEX for a benchmark scenario in the Oakland Air Route Traffic Control Center. Several computational aspects of the method are assessed-in particular, accuracy of the numerical discretization, computational time, and storage space required by the method.

It is sometimes necessary to consider more general nonlinear programs than the ones we have already studied: linear, quadratic, or conic programs. We give a brief introduction to this vast topic, and we discuss an application to estimating a volatility surface. Nonlinear Programming Consider a very general optimization problem of the form or the equivalent more concise form where. In the special case when all functions , hi are linear, problem (20.1) is a linear program as discussed in Chapter 2. When some of the functions , are nonlinear, problem (20.1) is a nonlinear program . Many practical problems are naturally formulated as nonlinear programs. We already saw quadratic programming and conic programming in earlier chapters. However, the family of problems that can be formulated as nonlinear programs is enormous, as many, if not most, imaginable kinds of constraints and objectives can be cast in terms of nonlinear functions. (For some noteworthy examples, see the exercises at the end of the chapter.) The immense modeling power of nonlinear programming comes at a cost: unlike linear, quadratic, and conic programming, which have a solid theory and are solvable via a few algorithmic templates, both the theory and methods to solve general nonlinear programs are far more complicated. Different types of nonlinear programs, determined by structural properties of the objective and constraint functions, are amenable to different types of algorithms. The subsequent sections sketch the main theory and most popular algorithmic ideas. A comprehensive treatment of this vast topic is beyond the scope of this textbook. We refer the reader to the excellent references Bertsekas (1999), Guler (2010), and Nocedal and Wright (2006) for more details. Numerical Nonlinear Programming Solvers There are numerous software packages for solving nonlinear programs. The following are some popular ones. We list them according to the class of algorithms (discussed in Section 20.4) that they are based on: (1) CONOPT, GRG2, Excel SOLVER. These solvers are based on the generalized reduced-gradient method . (2) MATLAB optimization toolbox, SNOPT, NLPQL. These solvers are based on sequential quadratic programming. (3) MINOS, LANCELOT. These solvers are based on an augmented Lagrangian approach. (4) MOSEK, LOQO, IPOPT. These solvers are based on interior-point methods.

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.

Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented and reconstruction algorithms based on linear programming and quadratic programming are derived. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions. Simulations for Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate.

We develop a single rounding algorithm for scheduling on unrelated parallel machines; this algorithm works well with the known linear programming, quadratic programming, and convex programming-relaxations for scheduling to minimize completion time, makespan, and other well-studied objective functions. We obtain the following applications for the general setting of unrelated parallel machines: (i) a bicriteria algorithm for a schedule whose weighted completion-time and makespan simultaneously exhibit the current-best individual approximations for these criteria (3/2 and 2, respectively); (ii) better-than-two approximation guarantees for scheduling under the L/sub p/ norm for all 1 < p < /spl infin/, improving on the 2-approximation algorithms of Azar & Epstein; and (iii) the first constant-factor multicriteria approximation algorithms that handle the weighted completion-time and any given collection of integer L/sub p/ norms. Our algorithm yields a common generalization of rounding theorems due to Karp et al and Shmoys & Tardos; among other applications, this yields an improved approximation for scheduling with resource-dependent processing times studied by Grigoriev et al.

Interregional and interseasonal competition in the U.S. beef industry have been studied extensively during the past decade. Linear programming, quadratic programming, simulation, and various other econometric models have formed the analytical frameworks in these studies.Reactive programming was first introduced as a useful tool in analyzing interregional competition problems by Tramel and Seale in 1959. Since that time, it has been utilized on a limited basis in spatial analyses in general, and the beef sector in particular. Goodwin used reactive programming in analyzing feeder cattle distribution patterns in the Southwest. In 1972, King and Ho reported a revised reactive programming algorithm and three illustrations of its applicability.