Primal Algorithm Research Articles

On the formulation of closest‐point projection algorithms in elastoplasticity—part II: Globally convergent schemes

AbstractThis paper presents the formulation of numerical algorithms for the solution of the closest‐point projection equations that appear in typical implementations of return mapping algorithms in elastoplasticity. The main motivation behind this work is to avoid the poor global convergence properties of a straight application of a Newton scheme in the solution of these equations, the so‐called Newton‐CPPM. The mathematical structure behind the closest‐point projection equations identified in Part I of this work delineates clearly different strategies for the successful solution of these equations. In particular, primal and dual closest‐point projection algorithms are proposed, in non‐augmented and augmented Lagrangian versions for the imposition of the consistency condition. The primal algorithms involve a direct solution of the original closest‐point projection equations, whereas the dual schemes involve a two‐level structure by which the original system of equations is staggered, with the imposition of the consistency condition driving alone the iterative process. Newton schemes in combination with appropriate line search strategies are considered, resulting in the desired asymptotically quadratic local rate of convergence and the sought global convergence character of the iterative schemes. These properties, together with the computational performance of the different schemes, are evaluated through representative numerical examples involving different models of finite‐strain plasticity. In particular, the avoidance of the large regions of no convergence in the trial state observed in the standard Newton‐CPPM is clearly illustrated. Copyright © 2001 John Wiley & Sons, Ltd.

Telecommunications Network Design—Comparison of Alternative Approaches

ABSTRACTThe design and development of the network infrastructure to support mission‐critical applications has become a critical and‐complex activity. This study explores the use of genetic algorithms (GA) for network design in the context of degree‐constrained minimal spanning tree (DCMST) problem; compares for small networks the performance of GA with a mathematical model that provides optimal solutions; and for larger networks, compares GA's performance with two heuristic methods—edge exchange and primal algorithm. Two performance measures, solution quality and computation time, are used for evaluation. The algorithms are evaluated on a wide variety of network sizes with both static and dynamic degree constraints on the network nodes. The results indicate that GA provides optimal solutions for small networks. For larger networks it provides better solution quality compared to edge exchange and primal method, but is worse than the two methods in computation time.

Interior-point linear programming algorithm for auction methods

Interior-point programming (IP) has been applied to many power system problems because of its efficiency for big problems. This paper illustrates application of interior-point linear programming (IPLP) to auction methods. An extended algorithm of IPLP is developed and used in this paper. This extended IPLP algorithm can find the exact optimal solution (i.e., exact optimal vertex) and can recover the optimal basis. Sensitivity analysis can be performed after the optimal basis is found. The sensitivity analysis performed in this paper is increase in the bid price and increase in the flow limit of the transmission line. This extended algorithm is expanded from the affine-scaling primal algorithm. The concept used in this extended algorithm to find the optimal vertex and optimal basis is simple.

Relaxed Most Negative Cycle and Most Positive Cut Canceling Algorithms for Minimum Cost Flow

This paper presents two new scaling algorithms for the minimum cost network flow problem, one a primal cycle canceling algorithm, the other a dual cut canceling algorithm. Both algorithms scale a relaxed optimality parameter, and create a second, inner relaxation. The primal algorithm uses the inner relaxation to cancel a most negative node-disjoint family of cycles w.r.t. the scaled parameter, the dual algorithm uses it to cancel most positive cuts w.r.t. the scaled parameter. We show that in a network with n nodes and m arcs, both algorithms need to cancel only O(mn) objects per scaling phase. Furthermore, we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms. Our algorithms have potential practical advantages compared to some other canceling algorithms as well. Along the way, we give a comprehensive survey of cycle and cut canceling algorithms for min-cost flow. We also clarify the formal duality between cycles and cuts.

On the numerical treatment of linearly constrained semi-infinite optimization problems

We consider the application of two primal algorithms to solve linear semi-infinite programming problems depending on a real parameter. Combining a simplex-type strategy with a feasible-direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently. The second algorithm runs a feasible-direction method first and then switches to the purification procedure. The linear programming subproblems that yield the search direction involve only a small subset of the constraints. These subsets are updated at each iteration using a multi-local optimization algorithm. Numerical test examples, taken from the literature in order to compare the numerical effort with other methods, show the efficiency of the proposed algorithms.

Price probing in the simplex method

This paper describes remote probing, a method that employs the principle of external pivoting and evaluates points with respect to their objective values from a given basic point, i.e., without ever moving out of that basic point. Once a point is deemed sufficiently good, the method moves to that point in a single simplex step, and from there on, the regular primal simplex method commences. In that sense, remote probing can be seen as preprocessing for the primal simplex algorithm.

Two optimization methods for complicated structures with multiparameter elements based on generalized condensation

A high-order condensation expression is proposed to generalize Duffin's condensation formula. Based on the higher-order condensation expression, two primal algorithms of the second-order are constructed to solve GGP (generalized geometric programming) problems. To overcome the difficulty of solving the structural optimization with multiparameter elements, two methods are proposed and implemented in a computer program. The first method takes one geometrical property of the cross-section as one component of the design variables for each element and adopts SQP (sequential quadratic programming) to solve the optimization model. The second method takes geometrical sizes of the cross-section as the design variables for each element and adopts primal algorithms of GGP problems to solve the optimization model.

Long-Step Primal Path-Following Algorithm for Monotone Variational Inequality Problems

In this paper, we present a long-step primal path-following algorithm and prove its global convergence under usual assumptions. It is seen that the short-step algorithm is a special case of the long-step algorithm for a specific selection of the parameters and the initial solution. Our theoretical result indicates that the long-step algorithm is more flexible. Numerical results indicate that the long-step algorithm converges faster than the short-step algorithm.

The Minimum Covering l_pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in R^n. A hypersphere is a setS(c,r)={x ∈ R^n : d(x,c) ≤ r} , where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l_2(x-c) . We consider the extension of this problem when d(x, c) is given by anyl_pb -norm, where 1 0,j=1,...,n , then S(c,r) is called anl_pb -hypersphere, in particular for p=2 and b_j=1, j=1,...,n, we obtain thel_2 -norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallestl_pb -hypersphere enclosing a large set of points inR^n .

The k-cardinality assignment problem

We consider a generalization of the assignment problem in which an integer k is given and one wants to assign k rows to k columns so that the sum of the corresponding costs is a minimum. The problem can be seen as a 2-matroid intersection, hence is solvable in polynomial time; immediate algorithms for it can be obtained from transformation to min-cost flow or from classical shortest augmenting path techniques. We introduce original preprocessing techniques for finding optimal solutions in which g⩽ k rows are assigned, for determining rows and columns which must be assigned in an optimal solution and for reducing the cost matrix. A specialized primal algorithm is finally presented. The average computational efficiency of the different approaches is evaluated through computational experiments.

Second-order method of generalized geometric programming for spatial frame optimization

To design concrete sectional sizes of beams, the optimization model of the spatial frame is transformed into a problem of GGP (generalized geometric programming) in terms of the Duffin's condensation formula. Adopting the strategy of two stages, the problem is solved by the structure stage and the elements stage, alternately. The first- and the second-order primal algorithm for the GGP, which are based on the first- and the second-order expansions under the logarithmic transformation in the logarithmic space, are applied in the elements stage and the structure stage, respectively. Numerical results including a large scale structure of the electrostatic precipitator show satisfactory efficiency of the method.

Outranking methods for multicriterion decision making: Arrow's and Raynaud's conjecture

Outranking methods constitute a class of ordinal ranking algorithms for multicriterion decision making. This paper is concerned with four such methods: Kohler’s primal and dual algorithms, and Arrow–Raynaud’s primal and dual algorithms. Arrow and Raynaud made the conjecture that these four methods yield the totality of “prudent orders” whenever the outranking matrix has the “constant sum” property. This paper shows that their conjecture is not valid.

Exploring Interior-Point Linear Programming Algorithms and Software.

Part 1 Preparations: modelling linear optimization problems the simplex algorithm a first look at an interior point algorithm. Part 2 Algorithms: the primal algorithm the dual algorithm the primal-dual algorithm implementation issues. Part 3 Solutions: the integrated environment command line operations.

The singly constrained assignment problem: An AP basis algorithm

This manuscript presents a specialized primal simplex algorithm to obtain near optimal integer solutions for the singly constrained assignment problem. An optimal solution for the continuous relaxation of the problem is obtained by generalizing the alternating basis algorithm of Barr, Glover, and Klingman for the pure assignment problem. Near optimal integer solutions are obtained by pivoting into the optimal basis the slack variable associated with the side constraint. Our empirical analysis indicated that for our test problems the soft-ware implementation of this algorithm was six times faster than CPLEX and four times faster than NETSIDE (a specialized code for network problems with side constraints). The integer solutions obtained in our tests were typically within 2% of the optimum of the continuous relaxation.

Operations Research: Deterministic Optimization Models.

1. Introduction. 2. Modeling Linear Programs. 3. Review of Matrix Algebra and Geometry. 4. Duality and Optimality Conditions in LP. 5. Hungarian Method: A Primal-Dual Method for the Assignment Problem. 6. Primal Algorithm for the Transportation Problem. 7. The Simplex Method for General LP. 8. Algorithms for Multiobjective Models. 9. Modeling Integer and Combinatorial Programs. 10. The Branch and Bound Approach. 11. Heuristic Methods for Combinatorial Optimization Problems. 12. Dynamic Programming. 13. Critical Path Methods in Project Management. 14. Nonlinear Programming. Index.

The expansion of functions under transformation and its application to optimization

This paper proposes the expansion of functions under transformation, which represents infinite kinds of extensions of Taylor's expansion. Based on this, Duffin's condensation formula is developed as a condensation formula of high order. Furthermore, a theorem of efficiently solving mathematical programming by using the expansion of functions is proposed. According to the condensation formula of high order, two kinds of second order primal algorithms are proposed for solving generalized geometric programming with rapid and stable convergence, which could be applied to a wide range of structural optimization problems. Finally, an efficient fully stressed design method under the function transformation is proposed using these expansions of functions.

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.

Anchoring Points and Cones of Opportunities in Interior Multiobjective Linear Programming

This paper presents a modification of one variant of Karmarkar's interior-point linear programming algorithm to Multiobjective Linear Programming (MOLP) problems. We show that by taking the variant known as the affine-scaling primal algorithm, generating locally-relevant scaling coefficients and applying them to the projected gradients produced by it, we can define what we refer to as anchoring points that then define cones in which we search for an optimal solution through interaction with the decision maker. Currently existing MOLP algorithms are simplex-based and make their progress toward the optimal solution by following the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an optimal solution may prove less sensitive to problem size. We refer to the class of MOLP algorithms resulting from this variant as Affine-Scaling Interior Multiobjective Linear Programming (ASIMOLP) algorithms.

A practical anti-degeneracy row selection technique in network linear programming

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.