A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables

This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modifications to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices.     

Application of MFCAV Riemann Solver to Maire's Cell-Centered Lagrangian Method

In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire's requirement, to the Maire's node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 (2007), 1781-1824]. In particular, we apply the so-called Multi-Fluid Channel on Averaged Volume (MFCAV) Riemann solver and a Riemann solver that adaptively combines the MFCAV solver with other more dissipative Riemann solvers to the Maire's scheme. It is noted that neither of the two solvers satisfies the Maire's requirement. Numerical experiments are presented to demonstrate that the application of the two Riemann solvers is successful.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

Coupled Multi-field and Multi-rate Problems – Numerical Solution and Stability Analysis

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis. Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Does an increase in the number of solvers benefit the seeker in heterogeneous contests?

In this paper, we analyze whether increasing the number of solvers benefits the seeker in heterogeneous contests. We show that if the number of solvers is large, the expected average quality (resp., expected maximum quality) of solutions is decreasing (resp., increasing). However, the results may not hold when the number of solvers is below a certain threshold. These imply that the seeker should adopt the restricted-entry policy or not based on different measure criteria and sizes of solvers.     

Adaptive use of iterative methods in predictor–corrector interior point methods for linear programming

In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. We summarize earlier results using a modified version of the OB1-R code of Lustig, Marsten, and Shanno, and we present results from a predictor–corrector code PCx modified to use adaptive iteration. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained.     

An Exact Algorithm for the Simplified Multiple Depot Crew Scheduling Problem

The Multiple Depot Crew Scheduling Problem (MD-CSP) appears in public transit systems (e.g., airline, bus and railway industry) and consists of determining the optimal duties for a set of crews (or vehicles) split among several depots in order to cover a set of timetabled trips satisfying a number of constraints. We consider the case in which every crew must return to the starting depot and limits are imposed on both the elapsed time and the working time of any duty. The MD-CSP is an extension of both the Multiple Depot Vehicle Scheduling Problem (MD-VSP) and the single depot Crew Scheduling Problem (CSP). The MD-CSP is formulated as a set partitioning problem with side constraints (SP), where each column corresponds to a feasible duty. In this paper we extend to the MD-CSP the exact method used by Bianco, Mingozzi and Ricciardelli (1994) for MD-VSP and that used by Mingozzi et al. (1999) for the CSP. We also introduce a new bounding procedure based on Lagrangian relaxation and column generation which can deal with the MD-CSP constraints. The computational results for both random and real-world test problems from the literature show that the new exact procedure outperforms, on the test problems used, other exact methods proposed in the literature for the MD-VSP and the CSP.     

On $L^2$-Stability Analysis of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

This paper is devoted to stability analysis of the acoustic wave equation exterior to a bounded scatterer, where the unbounded computational domain is truncated by the exact time-domain circular/spherical nonreflecting boundary condition (NRBC). Different from the usual energy method, we adopt an argument that leads to $L^2$-a priori estimates with minimum regularity requirement for the initial data and source term. This needs some delicate analysis of the involved NRBC. These results play an essential role in the error analysis of the interior solvers (e.g., finite-element/spectral- element/spectral methods) for the reduced scattering problems. We also apply the technique to analyze a time-domain waveguide problem.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the overlapping domain decomposition. We call such formulations substructured domain decomposition methods. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS. We show that RAS and SRAS are equivalent when used as iterative solvers, as they produce the same iterates, while they are substantially different when used as preconditioners for GMRES. We link the volume and substructured Krylov spaces and show that the iterates are different by deriving the least squares problems solved at each GMRES iteration. When used as iterative solvers, SRAS presents computational advantages over RAS, as it avoids computations with matrices and vectors at the volume level. When used as preconditioners, SRAS has the further advantage of allowing GMRES to store smaller vectors and perform orthogonalization in a lower dimensional space. We then consider nonlinear problems, and we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton’s method. In contrast to the linear case, we prove that Newton’s method applied to the preconditioned volume and substructured formulation produces the same iterates in the nonlinear case. Next, we introduce two-level versions of nonlinear SRAS and SRASPEN. Finally, we validate our theoretical results with numerical experiments.

     

A comprehensive comparison of metaheuristics for the repetition-free longest common subsequence problem

This paper deals with an NP-hard string problem from the bio-informatics field: the repetition-free longest common subsequence problem. This problem has enjoyed an increasing interest in recent years, which has resulted in the application of several pure as well as hybrid metaheuristics. However, the literature lacks a comprehensive comparison between those approaches. Moreover, it has been shown that general purpose integer linear programming solvers are very efficient for solving many of the problem instances that were used so far in the literature. Therefore, in this work we extend the available benchmark set, adding larger instances to which integer linear programming solvers cannot be applied anymore. Moreover, we provide a comprehensive comparison of the approaches found in the literature. Based on the results we propose a hybrid between two of the best methods which turns out to inherit the complementary strengths of both methods.     

Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations: Spectral analysis and computational results

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating PDEs in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear systems. Among the main objectives are the design of new, efficient iterative solvers and a rigorous analysis of their convergence speed. The link we have in mind is either the structure or the hidden structure that the involved coefficient matrices inherit, both from the continuous PDE and from the approximation scheme; in turn, the resulting structure is used for deducing spectral information, crucial for the conditioning and convergence analysis and for the design of more efficient solvers. As a specific problem, we consider the incompressible Navier–Stokes equations; as a numerical technique, we consider a novel family of high‐order, accurate discontinuous Galerkin methods on staggered meshes, and as tools, we use the theory of Toeplitz matrices generated by a function (in the most general block, the multilevel form) and the more recent theory of generalized locally Toeplitz matrix sequences. We arrive at a somehow complete picture of the spectral features of the underlying matrices, and this information is employed for giving a forecast of the convergence history of the conjugate gradient method, together with a discussion on new and more advanced techniques (involving preconditioning, multigrid, multi‐iterative solvers). Several numerical tests are provided and critically illustrated in order to show the validity and the potential of our analysis.     

Acceleration methods for the convergence of vector sequences applied to multi-physics problems

Partitioned solution approaches are well-known strategies to solve coupled problems and have demonstrated their applicability in a wide range of multi-physical simulations. Applying a partitioned approach opens up numerous possibilities, such as connecting different solvers or combining non-matching meshes and time-step sizes. On the other hand such a procedure has the drawback of only conditional stability, leading to poor convergence rates or - as a worst case scenario - to divergent behavior. In this contribution, we will give an interpretation of the iterative coupling process as a sequence of vectors converging to an unknown limit, which indeed represents a solution to the coupled problem. In order to improve computational efficiency, there are several practicable methods to accelerate the convergence of vector sequences. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simplex type method using multiple updatings for specially structured linear programs

Many important large-scale linear programs with special structures lead to special computational procedures which are more efficient than the ordinary procedure of the generalized methods. Problems and their solvers taking advantage of multiple updatings of the basis in the dual simplex type method are presented. Computational results run by the efficient algorithm and a standard code MINOS for the test problems are compared and analyzed. It is shown that the amount of work for the optimal solution for the problem can be reduced by the new algorithm.     

A modified active set algorithm for transportation discrete network design bi-level problem

Transportation discrete network design problem (DNDP) is about how to modify an existing network of roads and highways in order to improve its total system travel time, and the candidate road building or expansion plan can only be added as a whole. DNDP can be formulated into a bi-level problem with binary variables. An active set algorithm has been proposed to solve the bi-level discrete network design problem, while it made an assumption that the capacity increase and construction cost of each road are based on the number of lanes. This paper considers a more general case when the capacity increase and construction cost are specified for each candidate plan. This paper also uses numerical methods instead of solvers to solve each step, so it provides a more direct understanding and control of the algorithm and running procedure. By analyzing the differences and getting corresponding solving methods, a modified active set algorithm is proposed in the paper. In the implementation of the algorithm and the validation, we use binary numeral system and ternary numeral system to avoid too many layers of loop and save storage space. Numerical experiments show the correctness and efficiency of the proposed modified active set algorithm.     

Coupling methods for heat transfer and heat flow: Operator splitting and the parareal algorithm

We propose an operator splitting method for coupling heat transfer and heat flow equations. This work is motivated by the need to couple independent industrial heat transfer solvers (e.g., the Aura-Fluid software package) and heat flow solvers (e.g., Openfoam). Such packages are often used to simulate the influence of solar heat in car bodies and are coupled by A-B splitting techniques. The main goal of this work is the acceleration of the coupled software system by iterative operator splitting methods and additional time-parallelism using the parareal algorithm. We present these new splitting techniques along with some novel convergence results and test the splitting-parareal combination on various numerical problems.     

Structured regularization for barrier NLP solvers

Barrier methods have led to several nonlinear programming (NLP) solvers (e.g. IPOPT, KNITRO, LOQO). However, certain regularity conditions are required for convergence of these methods. These conditions are violated for optimization models with dependent constraints, thus leading to method failure. These shortcomings can be identified by checking the inertia of the KKT matrix, and current solvers either add regularizing terms to correct the inertia of the KKT matrix or revert to more expensive trust region methods to solve the barrier problem. This study improves on these approaches with a new structured regularization strategy; within the Newton step it identifies an independent subset of equality constraints and removes the remaining constraints without modifying the KKT matrix structure. This approach leads to more accurate Newton steps and faster convergence, while maintaining global convergence properties. Implemented in IPOPT with linear solvers HSL_MA57, HSL_MA97 and MUMPS, we present numerical experiments on hundreds of examples from the CUTEr test set, modified for dependency. These results show an average reduction in iterations of more than 50 % over the current version of IPOPT. In addition, several nonlinear blending problems are solved with the proposed algorithm, and improvements over existing regularization strategies are further demonstrated.     

Robust iterative methods for elliptic problems with highly varying coefficients in thin substructures

Summary. In this paper we introduce a class of robust multilevel interface solvers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to geometric decompositions by a tensor product of strongly non-uniform meshes. The global iterations convergence rate is shown to be of the order with respect to the number of degrees of freedom on the single subdomain boundaries, uniformly upon the coarse and fine mesh sizes, jumps in the coefficients and aspect ratios of substructures. As the first approach, we adapt the frequency filtering techniques [28] to construct robust smoothers on the highly non-uniform coarse grid. As an alternative, a multilevel averaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coarse grid preconditioner is shown to have (in a two level case) the condition number independent of the coarse mesh grading and jumps in the coefficients related to the coarsest refinement level. The proposed technique exhibited high serial and parallel performance in the skin diffusion processes modelling [20] where the high dimensional coarse mesh problem inherits a strong geometrical and coefficients anisotropy. The approach may be also applied to magnetostatics problems as well as in some composite materials simulation. Received December 27, 1994     

Bilevel optimization applied to strategic pricing in competitive electricity markets

In this paper, we present a bilevel programming formulation for the problem of strategic bidding under uncertainty in a wholesale energy market (WEM), where the economic remuneration of each generator depends on the ability of its own management to submit price and quantity bids. The leader of the bilevel problem consists of one among a group of competing generators and the follower is the electric system operator. The capability of the agent represented by the leader to affect the market price is considered by the model. We propose two solution approaches for this non-convex problem. The first one is a heuristic procedure whose efficiency is confirmed through comparisons with the optimal solutions for some instances of the problem. These optimal solutions are obtained by the second approach proposed, which consists of a mixed integer reformulation of the bilevel model. The heuristic proposed is also compared to standard solvers for nonlinearly constrained optimization problems. The application of the procedures is illustrated in case studies with configurations derived from the Brazilian power system.     

An integer linear formulation for the file transfer scheduling problem

In this paper, we propose a new integer linear programming (ILP) formulation for solving a file transfer scheduling problem (FTSP), which is to minimize the overall time needed to transfer all files to their destinations for a given collection of various sized files in a computer network. Each computer in this network has a limited number of communication ports. The described problem is proved to be NP-hard in a general case. Our formulation enables solving the problem by standard ILP solvers like CPLEX or Gurobi. To prove the validity of the proposed ILP formulation, two new reformulations of FTSP are presented. The results obtained by CPLEX and Gurobi solvers, based on this formulation, are presented and discussed.