Search

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS algorithm, the extended ABS algorithm, the block ABS algorithm, and the integer ABS algorithm were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, WZ, and ZW. We present the octant interlocking factorization and show that the generalized ABS algorithm is more general to produce the octant interlocking factorization. [ABSTRACT FROM AUTHOR]

We present a feasible full step interior-point algorithm to solve the P ∗ (κ) horizontal linear complementarity problem defined on a Cartesian product of symmetric cones, which is not based on a usual barrier function. The full steps are scaled utilizing the Nesterov-Todd (NT) scaling point. Our approach generates the search directions leading to the full-NT steps by algebraically transforming the centring equation of the system which defines the central trajectory using the induced barrier of a so-called positive-asymptotic kernel function. We establish the global convergence as well as a local quadratic rate of convergence of our proposed method. Finally, we demonstrate that our algorithm bears a complexity bound matching the best available one for the algorithms of its kind. [ABSTRACT FROM AUTHOR]

We present an improved version of a full Nesterov-Todd step infeasible interior-point method for linear complementarity problem over symmetric cone (Bull. Iranian Math. Soc., 40, (2014), no. 3, 541-564). In the earlier version, each iteration consisted of one so-called feasibility step and a few -at most three - centering steps. Here, each iteration consists of only a feasibility step. Thus, the new algorithm demands less work in each iteration and admits a simple analysis of complexity bound. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods. [ABSTRACT FROM AUTHOR]

We present an infeasible interior-point algorithm for symmetric linear complementarity problem based on modified Nesterov–Todd directions by using Euclidean Jordan algebras. The algorithm decreases the duality gap and the feasibility residual at the same rate. In this algorithm, we construct strictly feasible iterates for a sequence of perturbations of the given problem. Each main iteration of the algorithm consists of a feasibility step and a number of centring steps. The starting point in the first iteration is strictly feasible for a perturbed problem. The feasibility steps lead to a strictly feasible iterate for the next perturbed problem. By using centring steps for the new perturbed problem, a strictly feasible iterate is obtained to be close to the central path of the new perturbed problem. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods. [ABSTRACT FROM AUTHOR]

A convergent product is an assembly shape concept integrating functions and sub-functions to form a final product. To conceptualize the convergent product problem, a web-based network is considered in which a collection of base functions and sub-functions configure the nodes, and each arc in the network is considered to be a link between two nodes. The aim is to find an optimal tree of functionalities in the network, adding value to the product in the web environment. First, an algorithm is proposed to assign the links among bases and sub-functions. Then, numerical values, as benefits and costs, are determined for arcs and nodes, respectively, using a mathematical approach. Also, customer value corresponding to the benefits is considered. Finally, the Steiner tree methodology is adapted to a multi-objective model optimized by an augmented ε-constraint method. An example is worked out to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

A full Nesterov-Todd (NT) step infeasible interior-point algorithm is proposed for solving monotone linear complementarity problems over symmetric cones by using Euclidean Jordan algebra. Two types of full NT-steps are used, feasibility steps and centering steps. The algorithm starts from strictly feasible iterates of a per- turbed problem, and, using the central path and feasibility steps, finds strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, strictly fea- sible iterates are obtained to be close enough to the central path of the new perturbed problem. The starting point depends on two positive numbers ρp and ρd. The algorithm terminates either by finding an ∊-solution or detecting that the symmetric cone linear complementarity problem has no optimal solution with vanishing duality gap satisfying a condition in terms of ρp and ρd. The it- eration bound coincides with the best known bound for infeasible interior-point methods. [ABSTRACT FROM AUTHOR]

Classes of Abaffy-Broyden-Spedicato (ABS) methods have been introduced for solving linear systems of equations. The algorithms are powerful methods for developing matrix factoriza- tions and many fundamental numerical linear algebra processes. Here, we show how to apply the ABS algorithms to devise algo- rithms to compute the WZ and ZW factorizations of a nonsingular matrix as well as the WTW and ZTZ factorizations of a symmetric positives definite matrix. We also describe the QZ and the QW factorizations, with Q orthogonal, and show how to appropriate the parameters of the ABS algorithms to compute these factorizations. [ABSTRACT FROM AUTHOR]

We present a model and propose an approach to compute an approximate solution of Fully Fuzzy Linear System (FFLS) of equations in which all the components of the coefficient matrix are either nonnegative or nonpositive. First, in discussing an FFLS with a nonnegative coefficient matrix, we consider an equivalent FFLS by using an appropriate permutation to simplify fuzzy multiplications. To solve the m x n permutated system, we convert it to three m x n real linear systems, one being concerned with the cores and the other two being related to the left and right spreads. To decide whether the core system is consistent or not, we use the modified Huang algorithm of the class of ABS methods. If the core system is inconsistent, an appropriate unconstrained least squares problem is solved for an approximate solution. The sign of each component of the solution is decided by the sign of its core. Also, to know whether the left and right spread systems are consistent or not, we apply the modified Huang algorithm again. Appropriate constrained least squares problems are solved, when the spread systems are inconsistent or do not satisfy fuzziness conditions. Then, we consider the FFLS with a mixed single-signed coefficient matrix, in which each component of the coefficient matrix is either nonnegative or nonpositive. In this case, we break the m x n coefficient matrix up to two m x n matrices, one having only nonnegative and the other having only nonpositive components, such that their sum yields the original coefficient matrix. Using the distributive law, we convert each m x n FFLS into two real linear systems where the first one is related to the cores with size m x n and the other is 2m x 2n and is related to the spreads. Here, we also use the modified Huang algorithm to decide whether these systems are consistent or not. If the first system is inconsistent or the second system does not satisfy the fuzziness conditions, we find an approximate solution by solving a respective least squares problem. We summarize the proposed approach by presenting two computational algorithms. Finally, the algorithms are implemented and effectively tested by solving various randomly generated consistent as well as inconsistent numerical test problems. [ABSTRACT FROM AUTHOR]

We present a structured algorithm for solving constrained non-linear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected Hessians. We discuss the comparative results of the test-ing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and Mahdavi-Amiri. The results indeed confirm the practical significance of our special considerations for the inherent structure of the least squares. [ABSTRACT FROM AUTHOR]

We present an effective algorithm for minimization of locally nonconvex Lipschitz functions based on mollifier functions approximating the Clarke generalized gradient. To this aim, first we approximate the Clarke generalized gradient by mollifier subgradients. To construct this approximation, we use a set of averaged functions gradients. Then, we show that the convex hull of this set serves as a good approximation for the Clarke generalized gradient. Using this approximation of the Clarke generalized gradient, we establish an algorithm for minimization of locally Lipschitz functions. Based on mollifier subgradient approximation, we propose a dynamic algorithm for finding a direction satisfying the Armijo condition without needing many subgradient evaluations. We prove that the search direction procedure terminates after finitely many iterations and show how to reduce the objective function value in the obtained search direction. We also prove that the first order optimality conditions are satisfied for any accumulation point of the sequence constructed by the algorithm. Finally, we implement our algorithm with MATLAB codes and approximate averaged functions gradients by the Monte-Carlo method. The numerical results show that our algorithm is effectively more efficient and also more robust than the GS algorithm, currently perceived to be a competitive algorithm for minimization of nonconvex Lipschitz functions. [ABSTRACT FROM AUTHOR]

The article presents information on the Second International Conference on Operations Research and Optimization that was held on January 19-22, 2013, and organized by the School of Mathematics, Statistics, and Computer Science of the University of Tehran, Iran. Details related to the objective of this conference and the presented papers are included.