Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

In the first part of this work, we presented a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite \(\varepsilon \) -convergence to a global solution of the bilevel problem is proved. Thirty-four problems from the literature are tackled successfully.     

New geometric programming algorithms with numerical experiments

Geometric programming (GP) is one of the most important parts of mathematical programming. GP models are widely applied in different spheres of science, technology, national economy [1,2]. The paper presents six new algorithms of linear and polynomial approximation for the solution of GP programs. The results of extensive numerical experiments are also included in the paper. They indicate the effectiveness of the proposed methods.     

Convergence of a Dinkelbach-type algorithm in generalized fractional programming

The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.

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Zusammenfassung Die Konvergenz eines Verfahrens i. S. von Dinkelbach zur Lösung verallgemeinerter Quotientenprogramme wird durch Untersuchung der Sensitivität eines parametrisierten Problems abgeleitet. Es wird gezeigt, daß die Konvergenzrate durch (1+5)/2 nach unten beschränkt ist, falls gewisse Regularitätsbedingungen in einer Umgebung der Optimallösung erfüllt sind. Ferner wird eine notwendige und hinreichende Bedingung zur quadratischen Konvergenz hergeleitet. Es wird gezeigt, wie diese im Falle konvexer Probleme implementiert werden kann.

     

Computational results of an interior point algorithm for large scale linear programming

This paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming. The implementation uses the preconditioned conjugate gradient method for computing projections. Our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large LPs in which the direct methods fail due to memory and CPU time requirements. The conjugate gradient algorithm was able to find very accurate directions even when the system was ill-conditioned. The paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables. We have conducted extensive computational experiments on problems representative of large classes of applications of current interest. We have also chosen instances of the problems of future potential interest, which could not be solved in the past due to the weakness of the prior solution methods, but which represent a large class of new applications. The hypergraph model is such an example. Comparison of our implementation with MINOS 5.1 shows that our implementation is orders of magnitude faster than MINOS 5.1 for these problems.     

Convergence and numerical results for a parallel asynchronous quasi-Newton method

During the execution of a parallel asynchronous iterative algorithm, each task does not wait for predetermined data to become available. On the contrary, they can be viewed as local and independent iterative algorithms, which perform their own iterative scheme on the data currently available.On the basis of this computational model, a parallel asynchronous version of the quasi-Newton method for solving unconstrained optimization problems is proposed. The algorithm is based on four tasks concurrently executing and interacting in an asynchronous way.Convergence conditions are established and numerical results are presented which prove the effectiveness of the proposed parallel asynchronous approach.This research work was partially supported by the National Research Council of Italy within the special project Sistemi Informatici e Calcolo Parallelo under CNR Contract No. 92.01585.PF69.The authors are grateful to M. Al-Baali and R. H. Byrd for their valuable comments.     

Average case complexity results for a centering algorithm for linear programming problems under Gaussian distributions

To solve linear programming problems by interior point methods an approximately centered interior point has to be known. Such a point can be found by an algorithmic approach – a so-called phase 1 algorithm or centering algorithm. For random linear programming problems distributed according to the rotation symmetry model, especially with normal distribution, we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm.     

Convergence theorems of a hybrid algorithm for equilibrium problems

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of common fixed points of two hemi-relatively non-expansive mappings and the set of solutions of an equilibrium problem by the CQ hybrid method in Banach spaces. Our results improve and extend the corresponding results announced by Cheng and Tian [Y. Cheng, M. Tian, Strong convergence theorem by monotone hybrid algorithm for equilibrium problems, hemi-relatively nonexpansive mappings and maximal monotone operators, Fixed Point Theory Appl. 2008 (2008) 12 pages, doi:10.1155/2008/617248], Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively non-expansive mappings, Fixed Point Theory Appl. (2008) doi:10.1155/2008/528476] and some others.     

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkar's linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ⁿ having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ: ^{n m} to Karmarkar's original algorithm for a linear program in ^m havingm—n equality constraints andm inequality constraints. Karmarkar's original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkar's algorithm whose step size minimizes the potential function in the step direction.The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = (x) which maps the hyperplaneH _opt ={x:c, x =c ₀} specified by the optimal value of the objective function to the hyperplane at infinity. The feasible solution set is mapped under to anunbounded polytope. Letf _LB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkar's potential function in the new coordinate system. Theglobal Newton method iterate y ^* for a strictly convex functionf(y) defined on a suitable convex domain is that pointy ^* that minimizesf(y) on the search ray {y+ v _N(y): 0} wherev _N(y) =–(² f(y))^–1(f(y)) is the Newton's method vector. If {x ^(k)} are a set of projective scaling algorithm iterates in the original coordinate system andy ^(k) =(x ^(k)) then {y ^(k)} are a set of global Newton method iterates forf _LB(y) and conversely.Karmarkar's algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.     

A primal conical linear programming algorithm

This paper completes the treatment of the conical approach to linear programming, introducing a conical primal algorithm of linear programming. After some recalls and improvements of a previous paper dealing with such an approach, the algorithm is defined. A first convergence result is then proved, which, along with a series of lemmata, allows to prove that the algorithm terminates in a finite number of steps     

An efficient algorithm for linear programming

A simple but efficient algorithm is presented for linear programming. The algorithm computes the projection matrix exactly once throughout the computation unlike that of Karmarkar’s algorithm where in the projection matrix is computed at each and every iteration. The algorithm is best suitable to be implemented on a parallel architecture. Complexity of the algorithm is being studied.     

An algorithm for bicriterion linear programming

An algorithm is presented here for bicriterion linear programs which generates either all efficient points or a subset of such efficient points corresponding to a decision-maker's specified space of objective weights. The computational requirements of the algorithm are quite low; in fact only a series of divisions and comparisons are needed for the determination of adjacent efficient extreme points. As a by-product, the range of objective weights corresponding to each efficient extreme point is also generated. This additional information is used to characterize the set of all efficient points as a union of maximal efficient faces.     

A parallel integer linear programming algorithm

The solution of large scale integer linear programming models is generally dependent, in some way, upon the branch and bound technique, which can be quite time consuming. This paper describes a parallel branch and bound algorithm which achieves super linear efficiency in solving integer linear programming models on a multiprocessor computer. The algorithm is used to solve the Haldi and IBM test problems as well as a system design model.     

Global optimization algorithm for a generalized linear multiplicative programming

This article present a branch and bound algorithm for globally solving generalized linear multiplicative programming problems with coefficients. The main computation involve solving a sequence of linear relaxation programming problems, and the algorithm economizes the required computations by conducting the branch and bound search in R ^p , rather than in R ⁿ , where p is the number of rank and n is the dimension of decision variables. The proposed algorithm will be convergent to the global optimal solution by means of the subsequent solutions of the series of linear relaxation programming problems. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.     

Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems

We show that a particular pivoting algorithm, which we call the lexicographic Lemke algorithm, takes an expected number of steps that is bounded by a quadratic inn, when applied to a random linear complementarity problem of dimensionn. We present two probabilistic models, both requiring some nondegeneracy and sign-invariance properties. The second distribution is concerned with linear complementarity problems that arise from linear programming. In this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem, and independent of the larger. Similar results have been obtained by Adler and Megiddo.Research partially funded by a fellowship from the Alfred Sloan Foundation and by NSF Grant ECS82-15361.     

Exact algorithm for solving a special fixed-charge linear programming problem

A branch-and-bound algorithm (A) for solving a fixed-charge linear programming problem (P) involving identical fixed charges, one equality constraint, and explicit bounds on the variables is presented. Problem (P) can serve as a mathematical model for profit optimization in sawn timber production. Some theoretical considerations upon a fixed-charge problem (P), arising from (P) by permitting the fixed charges to be different for each variable, are carried out. A basic algorithm (A₀) is stated, and it is proved that Algorithm (A₀) finds an optimal solution of Problem (P) [resp., (P)] within a finite number of steps. Algorithm (A₀), combined with bounds developed with regard to Problem (P), yields Algorithm (A), which operates on a subset of all vertices of the feasible region. Finally, computational results concerning the numerical solution of Problem (P) by Algorithm (A) are stated.A part of this work was carried out in connection with the project Optimierung der Schnittholzproducktion auf Zerspaneranlagen, which was done at the Institute of Mathematics of the University of Klagenfurt in cooperation with the firm J. Offner, Holzindustrie GmbH, Wolfsberg. This project was partially supported by Forschungsförderungsfonds für die gewerbliche Wirtschaft. The author would like to thank Professor H. Stettner, C. Nowak, and H. Woschitz for their support and G. Stoiser for his help in achieving the numerical results.     

Finite algorithm for generalized linear multiplicative programming

The nonconvex problem of minimizing the sum of a linear function and the product of two linear functions over a convex polyhedron is considered. A finite algorithm is proposed which either finds a global optimum or shows that the objective function is unbounded from below in the feasible region. This is done by means of a sequence of primal and/or dual simplex iterations.The first author gratefully acknowledges the research support received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata all' Economia, Universitá di Pisa, Pisa, Italy, Spring 1992.     

A new polynomial-time algorithm for linear programming

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n ^3.5 L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n ^2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time. This is a substantially revised version of the paper presented at the Symposium on Theory of Computing, Washington D. C., April 1984.     

An infeasible-interior-point potential-reduction algorithm for linear programming

n . The method is based on Rockafellar’s proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p^a(x_k) of x_k to define a v_k∈∂_εkf(p^a(x_k)) with ε_k≤α∥v_k∥, where α is a constant. The method monitors the reduction in the value of ∥v_k∥ to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of ∥v_k∥. Without the differentiability of f, the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Moré. Numerical results show the good performance of the method. Received October 3, 1995 / Revised version received August 20, 1998 Published online January 20, 1999     

Khachiyan's linear programming algorithm

L. G. Khachiyan's polynomial time algorithm for determining whether a system of linear inequalities is satisfiable is presented together with a proof of its validity. The algorithm can be used to solve linear programs in polynomial time.     

A randomized algorithm for fixed-dimensional linear programming

We give a (Las Vegas) randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is , where lim _{d d} = 0. This improves the corresponding worst-case complexity, . The method is based on a recent idea of Clarkson. Two variations on the algorithm are examined briefly.