First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming

This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs with affine operator by using a continuously differentiable dual gap function. This extends an idea from Dietrich (J. Math. Anal. Appl. 235:380–393 [24]). Some numerical results illustrate the practical behavior of this dual gap function approach.     

A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation

A new quasi-Newton algorithm for the solution of general box constrained variational inequality problem (GVI(l, u, F, f)) is proposed in this paper. It is based on a reformulation of the variational inequality problem as a nonsmooth system of equations by using the median operator. Without smoothing approximation, the proposed quasi-Newton algorithm is directly applied to solve this class of nonsmooth equations. Under appropriate assumptions, it is proved that the algorithmic sequence globally and superlinearly converges to a solution of the equation reformulation and also of GVI(l, u, F, f). Numerical results show that our new algorithm works quite well.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

Dantzig–Wolfe decomposition of the daily course pattern formulation for curriculum-based course timetabling

In this paper, we considered the problem of Curriculum-Based Course Timetabling, i.e., assigning weekly lectures to a time schedule and rooms. We developed a Column Generation algorithm based on a pattern formulation of the time scheduling part of the problem by Bagger et al. (2016). The pattern formulation is an enumeration of all schedules by which each course can be assigned on each day; it is a lower bounding model. Pattern enumeration has also been considered in Burke (2008), where the authors enumerated all schedules to which each curriculum can be assigned on each day. We applied the Dantzig–Wolfe reformulation, so each column corresponded to a schedule for an entire day.We solved the reformulation with the Column Generation algorithm, where each pricing problem generated a full schedule for a single day. We provided a pre-processing technique that, on average, removed approximately 45% of the pattern variables in the pricing problems. We then extended the pre-processing technique into inequalities that we added to the model. Lastly, we describe how we applied Local Branching to the pricing problem by using the columns generated in previous iterations.We compare the lower bounds we obtained, with other methods from literature, on 20 data instances of real-world applications. For 16 instances the optimal solutions are known, but the remaining four are still open. Our approach improved the best-known lower bound for all four open instances, and decreased the average gap from 24 to 11%.     

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.     

On finite termination of an iterative method for linear complementarity problems

Based on a well-known reformulation of the linear complementarity problem (LCP) as a nondifferentiable system of nonlinear equations, a Newton-type method will be described for the solution of LCPs. Under certain assumptions, it will be shown that this method has a finite termination property, i.e., if an iterate is sufficiently close to a solution of LCP, the method finds this solution in one step. This result will be applied to a recently proposed algorithm by Harker and Pang in order to prove that their algorithm also has the finite termination property.     

Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems

In this paper we investigate a class of cardinality-constrained portfolio selection problems. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second-order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically constrained quadratic program (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effectiveness of the new MIQCQP reformulation.     

A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems

In this paper, we present a scenario aggregation algorithm for the solution of the dynamic minimax problem in stochastic programming. We consider the case where the joint probability distribution has a known finite support. The algorithm applies the Alternating Direction of Multipliers Method on a reformulation of the minimax problem using a double duality framework. The problem is solved by decomposition into scenario sub-problems, which are deterministic multi-period problems. Convergence properties are deduced from the Alternating Direction of Multipliers. The resulting algorithm can be seen as an extension of Rockafellar and Wets Progressive Hedging algorithm to the dynamic minimax context.     

Links Between Linear Bilevel and Mixed 0–1 Programming Problems

We study links between the linear bilevel and linear mixed 0–1 programming problems. A new reformulation of the linear mixed 0–1 programming problem into a linear bilevel programming one, which does not require the introduction of a large finite constant, is presented. We show that solving a linear mixed 0–1 problem by a classical branch-and-bound algorithm is equivalent in a strong sense to solving its bilevel reformulation by a bilevel branch-and-bound algorithm. The mixed 0–1 algorithm is embedded in the bilevel algorithm through the aforementioned reformulation; i.e., when applied to any mixed 0–1 instance and its bilevel reformulation, they generate sequences of subproblems which are identical via the reformulation.     

An improved Interior-Point Algorithm for Large-Scale Shakedown Analysis

This work focuses on the calculation of shakedown load-factors of structures by using the lower-bound theorem of shakedown analysis. The resultant optimization problem is solved by an interior-point algorithm. In general, the application of shakedown analysis to practical problems leads to large-scale problems with large numbers of unknowns and constraints. Thus the solution turns out to be computationally intensive. For this reason it is important to reduce the dimension of the considered problem. Therefore, the major improvement of the presented algorithm consists of a problem-oriented reformulation of the problem reducing the size of the system to be solved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

On the computation of protein backbones by using artificial backbones of hydrogens

NMR experiments provide information from which some of the distances between pairs of hydrogen atoms of a protein molecule can be estimated. Such distances can be exploited in order to identify the three-dimensional conformation of the molecule: this problem is known in the literature as the Molecular Distance Geometry Problem (MDGP). In this paper, we show how an artificial backbone of hydrogens can be defined which allows the reformulation of the MDGP as a combinatorial problem. This is done with the aim of solving the problem by the Branch and Prune (BP) algorithm, which is able to solve it efficiently. Moreover, we show how the real backbone of a protein conformation can be computed by using the coordinates of the hydrogens found by the BP algorithm. Formal proofs of the presented results are provided, as well as computational experiences on a set of instances whose size ranges from 60 to 6000 atoms.     

An Artificial Allocations Based Solution to the Label Switching Problem in Bayesian Analysis of Mixtures of Distributions

Label switching is a well-known problem occurring in MCMC outputs in Bayesian mixture modeling. In this article we propose a formal solution to this problem by considering the space of the artificial allocation variables. We show that there exist certain subsets of the allocation space leading to a class of nonsymmetric distributions that have the same support with the symmetric posterior distribution and can reproduce it by simply permuting the labels. Moreover, we select one of these distributions as a solution to the label switching problem using the simple matching distance between the artificial allocation variables. The proposed algorithm can be used in any mixture model and its computational cost depends on the length of the simulated chain but not on the parameter space dimension. Real and simulated data examples are provided in both univariate and multivariate settings. Supplemental material for this article is available online.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

Functional aspects of the Hardy inequality: appearance of a hidden energy

We revisit the classical Hardy inequality and the evolution problem associated to its Euler-Lagrange elliptic operator in the spirit of the paper by Vázquez and Zuazua (J Funct Analysis 173:103?C153 2000). As a result, we establish the correct optimal functional setting by means of a reformulation of the problem that eliminates the possible indefinite character of the Hardy functional due to the singular behaviour of solutions near the origin. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem already arises when the equation is posed in a bounded domain, a case we study in full detail. We also consider an equivalent problem with inverse-square potential on an exterior domain. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.     

On the Solution of Generalized Multiplicative Extremum Problems

The paper addresses the problem of maximizing a sum of products of positive and concave real-valued functions over a convex feasible set. A reformulation based on the image of the feasible set through the vector-valued function which describes the problem, combined with an adequate application of convex analysis results, lead to an equivalent indefinite quadratic extremum problem with infinitely many linear constraints. Special properties of this later problem allow to solve it by an efficient relaxation algorithm. Some numerical tests illustrate the approach proposed.     

One-level reformulation of the bilevel Knapsack problem using dynamic programming

In this paper, we consider the Bilevel Knapsack Problem (BKP), which is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions for a parametric Knapsack Problem. We introduce a new reformulation of the BKP into a one-level integer programming problem using dynamic programming. We propose an algorithm that allows the BKP to be solved exactly in two steps. In the first step, a dynamic programming algorithm is used to compute the set of follower reactions to leader decisions. In the second step, an integer problem that is equivalent to the BKP is solved using a branch-and-bound algorithm. Numerical results are presented to show the performance of our method.