The tree and forest spaces with applications to initial-value problem methods

Tree and forest spaces, which are at the heart of the theory of Runge–Kutta methods, are formulated recursively, and it is shown that the forest space is an algebra. To obtain order conditions in a systematic manner, Banach algebras are introduced to generate both the elementary weights for a general Runge–Kutta method and the corresponding quantities based on the Picard integral. To connect these two concepts, the Picard integral is written as the limiting case of an s-stage Runge–Kutta method, equivalent to s steps of the Euler method, as s tends to infinity. This approach makes it possible to make direct use of the tree space without going over to the dual space. By choosing linear combinations of trees, appropriate to a particular application, it is shown how to obtain alternative ways of writing the order conditions. This leads to a simpler and more direct derivation of particular methods.     

Risk preference modeling with conditional average: an application to portfolio optimization

The paper introduces a new risk measure called Conditional Average (CAVG), which was designed to cover typical attitudes towards risk for any type of distribution. It can be viewed as a generalization of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), two commonly used risk measures. The preference structure induced by CAVG has the interpretation in Yaari’s dual theory of choice under risk and relates to Tversky and Kahneman’s cumulative prospect theory. The measure is based on the new stochastic ordering called dual prospect stochastic dominance, which can be considered as a dual stochastic ordering to recently developed prospect stochastic dominance. In general, CAVG translates into a nonconvex quadratic programming problem, but in the case of a finite probability space it can also be expressed as a mixed-integer program. The paper also presents the results of computational studies designed to assess the preference modeling capabilities of the measure. The experimental analysis was performed on the asset allocation problem built on historical values of S&P 500 sub-industry indexes. The research was supported by the grant PBZ-KBN-016/P03/99 from the State Committee for Scientific Research.     

A multiobjective model for forest planning with adjacency constraints

This work presents a nonlinear goal programming model with binary variables used to plan the management of a tree plantation, taking economic and environmental objectives into account. The aims are to stay within the limits of a given harvesting volume, limit the age of basic units targeted for clearcutting, obtain a forest with a balanced age distribution, and surpass the minimum net present value set at each planning period. This has to be achieved bearing in mind technical restrictions regarding treatments, and spatial adjacency constraints that limit the maximum adjacent surface area to which clearcutting can be applied. The outcome is a highly complex problem that is solved by applying a metaheuristic method based on Scatter Search. The proposed model has been validated by applying it to a Cuban plantation located in the region of Pinar del Río.     

Existence results for second-order system with impulse effects via variational methods

In this paper, we investigate the positive solutions of a second-order system with impulse effects. By using critical point theory the existence result of positive solutions is obtained.     

A DEA methodology to evaluate the impact of information asymmetry on the efficiency of not-for-profit organizations with an application to higher education in Brazil

This paper presents a conceptual framework and an analytical DEA model for evaluating the impact of information asymmetry on organizational efficiency. The framework uses concepts from agency theory to estimate the extent of moral hazard by comparing the objectives of the principal to those of the agent. The framework and model are useful in the analysis of both for-profit and not-for-profit organizations because DEA is applicable whether or not inputs and/or outputs are subject to pricing mechanisms. An illustration focusing on the Brazilian not-for-profit federal university system finds that the agency problem indeed exists for a subset of those institutions, indicating the desirability of improved incentive and control mechanisms on the part of the principal.     

Moment calculations for piecewise-defined functions: an application to stochastic optimization with coherent risk measures

This work introduces a new analytical approach to the formulation of optimization problems with piecewise-defined (PD) objective functions. First, we introduce a new definition of multivariate PD functions and derive formal results for their continuity and differentiability. Then, we obtain closed-form expressions for the calculation of their moments. We apply these findings to three classes of optimization problems involving coherent risk measures. The method enables one to obtain insights on problem structure and on sensitivity to imprecision at the problem formulation stage, eliminating reliance on ad-hoc post-optimality numerical calculations.     

Malmquist indexes with quasi-fixed inputs: an application to school districts in Québec

We present a space and time allocation problem that arises in assembly halls producing large building blocks (namely, a shipyard which assembles prefabricated keel elements). The building blocks are very large, and, once a block is placed in the hall, it cannot be moved until all assembly operations on this block are complete. Each block must be processed during a predetermined time window. The objective is to maximize the number of building blocks produced in the hall.     

Solving a tactical operating room planning problem by a column-generation-based heuristic procedure with four criteria

In some hospitals, an “open scheduling” strategy is applied to solve the operating room planning problem; i.e., surgeons can choose any workday for his surgical cases, and the staffing of anesthetists and nurses is adjusted to maximize the efficiency of operating room utilization. In this paper, we aim at obtaining an efficient operating program for an operating theatre with several multifunctional operating rooms by using this “open scheduling” strategy. First, a mathematical model is constructed to assign surgical cases to operating rooms within one week. This model complies with the availability of operating rooms and surgeons, and its objective is not only to maximize utilization of operating rooms, but to minimize their overtime cost. Then a column-generation-based heuristic (CGBH) procedure is proposed, where four different criteria are compared with each other so as to find a solution with the best performance. In addition, the best approximate solution, obtained by this CGBH procedure after running all the criteria proposed, is compared with the lower bound obtained by an explicit column generation (CG) procedure, LP, to evaluate the distance between the approximate solution obtained and the optimum one. Although no criterion, according to the experimental results, is found superior to all other three in both robustness and quality of the solution obtained, it is found that the best solution obtained among those four criteria is often very close to LP, which means that the proposed algorithm can obtain a near optimal solution. In one word, the CGBH procedure proposed in this paper can obtain an efficient assignment of the surgical cases if the other resources (anesthesia and nursing staff, equipment, beds in the recovery room and etc.) are well organized.     

The effective application of a new approach to the generalized orienteering problem

The Orienteering Problem (OP) is an important problem in network optimization in which each city in a network is assigned a score and a maximum-score path from a designated start city to a designated end city is sought that is shorter than a pre-specified length limit. The Generalized Orienteering Problem (GOP) is a generalized version of the OP in which each city is assigned a number of scores for different attributes and the overall function to optimize is a function of these attribute scores. In this paper, the function used was a non-linear combination of attribute scores, making the problem difficult to solve. The GOP has a number of applications, largely in the field of routing. We designed a two-parameter iterative algorithm for the GOP, and computational experiments suggest that this algorithm performs as well as or better than other heuristics for the GOP in terms of solution quality while running faster. Further computational experiments suggest that our algorithm also outperforms the leading algorithm for solving the OP in terms of solution quality while maintaining a comparable solution speed.     

A MIP-based framework and its application on a lot sizing problem with setup carryover

In this paper we present a framework to tackle mixed integer programming problems based upon a “constrained” black box approach. Given a MIP formulation, a black-box solver, and a set of incumbent solutions, we iteratively build corridors around such solutions by adding exogenous constraints to the original MIP formulation. Such corridors, or neighborhoods, are then explored, possibly to optimality, with a standard MIP solver. An iterative approach in the spirit of a hill climbing scheme is thus used to explore subportions of the solution space. While the exploration of the corridor relies on a standard MIP solver, the way in which such corridors are built around the incumbent solutions is influenced by a set of factors, such as the distance metric adopted, or the type of method used to explore the neighborhood. The proposed framework has been tested on a challenging variation of the lot sizing problem, the multi-level lot sizing problem with setups and carryovers. When tested on 1920 benchmark instances of such problem, the algorithm was able to solve to near optimality every instance of the benchmark library and, on the most challenging instances, was able to find high quality solutions very early in the search process. The algorithm was effective, in terms of solution quality as well as computational time, when compared with a commercial MIP solver and the best algorithm from the literature.     

Block relaxation and majorization methods for the nearest correlation matrix with factor structure

We propose two numerical methods, namely the alternating block relaxation method and the alternating majorization method, for the problem of nearest correlation matrix with factor structure, which is highly nonconvex. In the block relaxation method, the subproblem is of the standard trust region problem, which is solved by Steighaug’s truncated conjugate gradient method or by the exact trust region method. In the majorization method, the subproblem has a closed-form solution. We then apply the majorization method to the case where nonnegative factors are required. The numerical results confirm that the proposed methods work quite well and are competitive against the best available methods.     

Erratum to: Risk-Sensitive Control with Near Monotone Cost

The infinite horizon risk-sensitive control problem for non-degenerate controlled diffusions is analyzed under a ‘near monotonicity’ condition on the running cost that penalizes large excursions of the process.     

Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrödinger type equation on torus

We consider the time local and global well-posedness for the fourth order nonlinear Schrödinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).     

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both static (buckling) and dynamic (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions, we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires.     

Exact penalties for variational inequalities with applications to nonlinear complementarity problems

In this paper, we present a new reformulation of the KKT system associated to a variational inequality as a semismooth equation. The reformulation is derived from the concept of differentiable exact penalties for nonlinear programming. The best theoretical results are presented for nonlinear complementarity problems, where simple, verifiable, conditions ensure that the penalty is exact. We close the paper with some preliminary computational tests on the use of a semismooth Newton method to solve the equation derived from the new reformulation. We also compare its performance with the Newton method applied to classical reformulations based on the Fischer-Burmeister function and on the minimum. The new reformulation combines the best features of the classical ones, being as easy to solve as the reformulation that uses the Fischer-Burmeister function while requiring as few Newton steps as the one that is based on the minimum.     

Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization

We propose and analyze a perturbed version of the classical Josephy–Newton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods. For the linearly constrained Lagrangian methods, in particular, we obtain superlinear convergence under the second-order sufficient optimality condition and the strict Mangasarian–Fromovitz constraint qualification, while previous results in the literature assume (in addition to second-order sufficiency) the stronger linear independence constraint qualification as well as the strict complementarity condition. For the sequential quadratically constrained quadratic programming methods, we prove primal-dual superlinear/quadratic convergence under the same assumptions as above, which also gives a new result.     

Supply network planning in the forest supply chain with bucking decisions anticipation

We consider a two-echelon timber supply chain in which the first echelon consists of several stands to be harvested and the second echelon consists of mills to be supplied with logs of different length. This problem aims at minimizing harvesting and transportation costs for one production period, while satisfying demand expressed as a mix of volumes of specific log types. Harvesting cost, which includes felling, bucking and hauling to roadside, depends upon the number of log type to be produced and sorted. Each stand to be harvested is modeled individually with a limited number of trees of various classes of diameter and total length, which affects the productivity factors of the bucking patterns to be used. To take these characteristics into account, we propose heuristics based on columns generation to solve the supply network problem at the forest level with an anticipation of bucking operations at the stand level.     

The normal ΔH-matrices with connection to some Jacobi-like methods

The ΔH-matrices appear in the context of certain maximization and minimization problems. The following purely algebraic problem, arising in connection with certain norm-reducing Jacobi-type algorithms is treated: When is a normal ΔH-matrix diagonal? Final results are obtained in the cases n = 3 and n = 4, and in general for the attractive normal ΔH-matrices.     

Metaheuristic methods based on Tabu search for assigning judges to competitions

Two metaheuristic methods based on Tabu search are introduced to assign judges to individual competitions in a tournament. The complexity of the mathematical formulation accounting for the assignment rules, leads us to use such an approach. The first metaheuristic includes two different Tabu searches that are combined with a diversification strategy. The second metaheuristic is applied to a penalized version of the original model formulated as an assignment problem. This metaheuristic is also based on a Tabu search procedure including a diversification strategy driven by the constraints violated. Numerical results are provided to indicate the efficiency of the methods to generate very good solutions.