Parameters Of Problem Research Articles

Sensitivity Analysis of Variational Inequalities via Twice Epi-differentiability and Proto-differentiability of the Proximity Operator

In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space. The challenge of the present work is to take into account a perturbation on all the data of the problem. This requires special adjustments in the definitions of the generalized first- and second-order differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly studied by R.T. Rockafellar, of twice epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous convex function and its subdifferential respectively. The link between these two notions is tied to Attouch's theorem and to the new concept, introduced in this paper, of convergent supporting hyperplanes. The previous tools allow us to derive an exact formula of the proto-derivative of the generalized proximity operator associated to a parameterized variational inequality, and deduce the differentiability of the associated solution with respect to the parameter. Furthermore, the derivative is shown to be the solution of a new variational inequality involving semi- and second epi-derivatives of the data. An application is given to parameterized convex optimization problems involving the sum of two convex functions (one of them being smooth). The case of smooth convex optimization problems with inequality constraints is discussed in details. This approach seems to be new in the literature and open several perspectives towards theoretical and computational issues in nonlinear optimization.

Fractals for Kernelization Lower Bounds

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim., 8 (2011), pp. 77--86], we show that, unless ${NP}\subseteq {{coNP}}/{{poly}}$, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most $k$ edges such that the resulting graph has no $s$-$t$ path of length shorter than $\ell$) parameterized by the combination of $k$ and $\ell$ has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex-deletion problems. Along the way, we show that LBEC remains NP-hard on planar graphs, a result which we believe is interesting in its own right.

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

Abstract We analyze the numerical performance of a preconditioning technique recently proposed in [1] for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coefficients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.

Parameterized LMI Based Diagonal Dominance Compensator Study for Polynomial Linear Parameter Varying System

For dynamic decoupling of polynomial linear parameter varying(PLPV) system, a robust dominance pre-compensator design method is given. The parameterized precompensator design problem is converted into an optimal problem constrained with parameterized linear matrix inequalities(PLMI) by using the conception of parameterized Lyapunov function(PLF). To solve the PLMI constrained optimal problem, the precompensator design problem is reduced into a normal convex optimization problem with normal linear matrix inequalities (LMI) constraints on a new constructed convex polyhedron. Moreover, a parameter scheduling pre-compensator is achieved, which satisfies robust performance and decoupling performances. Finally, the feasibility and validity of the robust diagonal dominance pre-compensator design method are verified by the numerical simulation on a turbofan engine PLPV model.

Parameterized approximation via fidelity preserving transformations

We motivate and describe a new parameterized approximation paradigm which studies the interaction between approximation ratio and running time for any parametrization of a given optimization problem. As a key tool, we introduce the concept of an α-shrinking transformation, for α≥1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving the approximation ratio of α (or α-fidelity). Moving even beyond the approximation ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain approximative kernels which are smaller than the best known for a given problem. The smaller “α-fidelity” kernels allow us to obtain an exact solution for the reduced instance more efficiently, while obtaining an approximate solution for the original instance. We show that such fidelity preserving transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree.

Handling time-varying constraints and objectives in dynamic evolutionary multi-objective optimization

Abstract Recently, several researchers within the evolutionary and swarm computing community have been interested in solving dynamic multi-objective problems where the objective functions, the problem's parameters, and/or the constraints may change over time. According to the related literature, most works have focused on the dynamicity of objective functions, which is insufficient since also constraints may change over time along with the objectives. For instance, a feasible solution could become infeasible after a change occurrence, and vice versa. Besides, a non-dominated solution may become dominated, and vice versa. Motivated by these observations, we devote this paper to focus on the dynamicity of both: (1) problem's constraints and (2) objective functions. To achieve our goal, we propose a new self-adaptive penalty function and a new feasibility driven strategy that are embedded within the NSGA-II and that are applied whenever a change is detected. The feasibility driven strategy is able to guide the search towards the new feasible directions according to the environment changes. The empirical results have shown that our proposal is able to handle various challenges raised by the problematic of dynamic constrained multi-objective optimization. Moreover, we have compared our new dynamic constrained NSGA-II version, denoted as DC-MOEA, against two existent dynamic constrained evolutionary algorithms. The obtained results have demonstrated the competitiveness and the superiority of our algorithm on both aspects of convergence and diversity.

Dealing with several parameterized problems by random methods

In this paper, we apply random methods to deal with several parameterized problems. For the Parameterized Weighted P3-Packing problem, by randomly partitioning the vertices in given graph, a tripartite graph can be obtained. We prove that the Parameterized Weighted P3-Packing problem can be solved in polynomial time on tripartite graphs. Based on the algorithm on tripartite graphs, a randomized parameterized algorithm of running time O⁎(32k) is given for the Parameterized Weighted P3-Packing problem. For the Parameterized Weighted Load Coloring problem, by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts, a randomized parameterized algorithm of running time O⁎(11.32k) is presented. For the Parameterized Claw-free Edge Deletion problem on Diamond-free Graphs, by combining random with branching methods, a parameterized algorithm of running time O⁎(2.42k) is given.

Fixed-point based hierarchical MPC control design for a cryogenic refrigerator

In this paper, a simple, general and scalable hierarchical control framework is proposed and validated through the interconnection of the Joule–Thomson and the Brayton cycle stages of a cryogenic refrigerator. The proposed framework enables to handle the case of destabilizing interconnections through state and/or control signals (which is the case of the cryogenic refrigerator example). Moreover, it offers the possibility to simply change the behavior of the overall system (depending on the context) by only changing the coordinator problem's parameters without changing the set of local controllers used by subsystems which is a common industrial requirement regarding industrial control architectures. Finally, the proposed scheme enables a smooth operator handover on a specific subsystem and/or actuator.

Train Timetabling on double track and multiple station capacity railway with useful upper and lower bounds

Train scheduling is one of the significant issues in the railway industry in recent years since it has an important role in efficacy of railway infrastructure. In this paper, the timetabling problem of a multiple tracked railway network is discussed. More specifically, a general model is presented here in which a set of operational and safety requirements is considered. The model handles the trains overtaking in stations and considers the stations capacity. The objective function is to minimize the total travel time. Unfortunately, the problem is NP-hard and real size problems cannot be solved in an acceptable amount of time. In order to reduce the processing time, we presented some heuristic rules, which reduce the number of binary variables. These rules are based on problem's parameters such as travel time, dwell time and safety time of stations and try to remove the impracticable areas of the solution space. Furthermore, a Lagrangian Relaxation algorithm model is presented in order to find a lower-bound. Finally, comprehensive numerical experiments on the Tehran Metro case are reported. Results show the efficiency of the heuristic rules and also the Lagrangian Relaxation method in a way that for all analyzed problems the optimum value are obtained.

Modeling of composite plates with an arbitrary hole location using the variable separation method

In this paper, a method to compute explicit solutions for laminated plate with arbitrary hole position is presented using a variable separation method. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x,y, the transverse coordinate z and the coordinates XT,YT of the hole position. As the parameterized problem involves the geometry, a mapping transformation is introduced to refer to a fixed reference configuration. This choice yields to an iterative process that consists of solving a 2D and three 1D problems successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element Method is used. The functions of XT and YT are discretized with linear interpolations. Mechanical tests with different numbers of layers are performed to show the accuracy of the method and its capacity to capture local effects and 3D state of the stress near the curved free-edge.

Fréchet Distance Between a Line and Avatar Point Set

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes $$\mathcal {Q}=\{Q_1,Q_2,\ldots ,Q_n\}$$ , where each class $$Q_i$$ consists of a pair of points $$Q_i = \{q_i, \bar{q_i}\}$$ . We call a subset $$A\subset \{q_i , \bar{q_i}:1\le i\le n\}$$ conflict-free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict-free subset that optimizes a given function. Given a line-segment $$\ell $$ and a set $$\mathcal {Q}$$ of a pair of points in $$\mathbb {R}^2$$ , our objective is to find a conflict-free subset that minimizes the Frechet distance between $$\ell $$ and the point set, where the minimum is taken over all possible conflict-free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm for the problem when parametrized by the solution size, which is later derandomized using universal family of sets. We believe that our derandomization technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in $$\mathcal {O}(2^k n \log ^2 n)$$ time, and the derandomized deterministic algorithm runs in $$2^k k^{\mathcal {O}(\log k)} n \log ^2 n$$ time, where k, the parameter, is the number of elements in the conflict-free subset solution. Finally we present a simple branching algorithm for the problem running in $$\mathcal {O}(2^k n\log n)$$ time. We also show that the problem does not have a polynomial sized kernel under standard complexity theoretic assumptions.

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

Kernelization is a strong and widely applied technique in parameterized complexity. In a nutshell, a kernelization algorithm for a parameterized problem transforms in polynomial time a given instance of the problem into an equivalent instance whose size depends solely on the parameter. Recent years have seen major advances in the study of both upper and lower bound techniques for kernelization, and by now this area has become one of the major research threads in parameterized complexity. In this article, we consider kernelization for problems on d -degenerate graphs, that is, graphs such that any subgraph contains a vertex of degree at most d . This graph class generalizes many classes of graphs for which effective kernelization is known to exist, for example, planar graphs, H -minor free graphs, and H -topological-minor free graphs. We show that for several natural problems on d -degenerate graphs the best-known kernelization upper bounds are essentially tight. In particular, using intricate constructions of weak compositions, we prove that unless coNP ⊆ NP/poly: • D ominating S et has no kernels of size O ( k ( d −1)( d −3)−ε ) for any ε > 0. The current best upper bound is O ( k (d+1) 2 ). • I ndependent D ominating S et has no kernels of size O ( k d −4−ε ) for any ε > 0. The current best upper bound is O ( k d +1 ). • I nduced M atching has no kernels of size O ( k d −3−ε ) for any ε > 0. The current best upper bound is O ( k d ). To the best of our knowledge, D ominating S et is the the first problem where a lower bound with superlinear dependence on d (in the exponent) can be proved. In the last section of the article, we also give simple kernels for C onnected V ertex C over and C apacitated V ertex C over of size O ( k d ) and O ( k d +1 ), respectively. We show that the latter problem has no kernels of size O ( k d −ε ) unless coNP ⊆ NP/poly by a simple reduction from d -E xact S et C over (the same lower bound for C onnected V ertex C over on d -degenerate graphs is already known).

Geometric optimal control and applications to aerospace

This article deals with applications of optimal control to aerospace problems with a focus on modern geometric optimal control tools and numerical continuation techniques. Geometric optimal control is a theory combining optimal control with various concepts of differential geometry. The ultimate objective is to derive optimal synthesis results for general classes of control systems. Continuation or homotopy methods consist in solving a series of parameterized problems, starting from a simple one to end up by continuous deformation with the initial problem. They help overcoming the difficult initialization issues of the shooting method. The combination of geometric control and homotopy methods improves the traditional techniques of optimal control theory.A nonacademic example of optimal attitude-trajectory control of (classical and airborne) launch vehicles, treated in details, illustrates how geometric optimal control can be used to analyze finely the structure of the extremals. This theoretical analysis helps building an efficient numerical solution procedure combining shooting methods and numerical continuation. Chattering is also analyzed and it is shown how to deal with this issue in practice.

New classes of parameterized differential Galois groups

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups.

A multivariate framework for weighted FPT algorithms

We introduce a multivariate approach for solving weighted parameterized problems. By allowing flexible use of parameters, our approach defines a framework for applying the classic bounded search trees technique. In our model, given an instance of size n of a minimization/maximization problem, and a parameter W≥1, we seek a solution of weight at most/at least W. We demonstrate the usefulness of our approach in solving Vertex Cover, 3-Hitting Set, Edge Dominating Set and Max Internal Out-Branching. While the best known algorithms for these problems admit running times of the form cWnO(1), for some c>1, our framework yields running times of the form csnO(1), where s≤W is the minimum size of a solution of weight at most/at least W. If no such solution exists, s=min⁡{W,m}, where m is the maximum size of a solution. In addition, we analyze the parameter t≤s, the minimum size of a solution.

Parameterized complexity classes beyond para-NP

Today's propositional satisfiability (SAT) solvers are extremely powerful and can be used as an efficient back-end for solving NP-complete problems. However, many fundamental problems in logic, in knowledge representation and reasoning, and in artificial intelligence are located at the second level of the Polynomial Hierarchy or even higher, and hence for these problems polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. Recent research shows that in certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions to SAT which exploit structural aspects of problem instances in terms of problem parameters. These reductions are more powerful because their running times can grow superpolynomially in the problem parameters. In this paper we develop a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not.

Estimation of quantiles of non-stationary demand distributions

ABSTRACTMany problems involve the use of quantiles of the probability distributions of the problem's parameters. A well-known example is the newsvendor problem, where the optimal order quantity equals a quantile of the demand distribution function. In real-life situations, however, the demand distribution is usually unknown and has to be estimated from past data. In these cases, quantile prediction is a complicated task, given that (i) the number of available samples is usually small and (ii) the demand distribution is not necessarily stationary. In some cases the distribution type can be meaningfully presumed, whereas the parameters of the distribution remain unknown. This article suggests a new method for estimating a quantile at a future time period. The method attaches weights to the available samples based on their chronological order and then, similar to the sample quantile method, it sets the estimator at the sample that reaches the desired quantile value. The method looks for the weights that minimize the expected absolute error of the estimator. A method for determining optimal weights in both stationary and non-stationary settings of the problem is developed. The applicability of the method is illustrated by solving a problem that has limited information regarding the distribution parameters and stationarity.

Representation of hypergeometric products in difference rings

In his pioneering work [1, 2], Michael Karr introduced ΠΣ-fields which provide a rather general framework for symbolic summation. He worked out the first algorithmic steps to represent indefinite nested sums and products as transcendental extensions over a computable ground field K called the field of constants. Furthermore, he presented an algorithm that solves the parameterized telescoping problem, and as special cases the telescoping and creative telescoping problems [3] within a given ΠΣ-field.

Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems

The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

AbstractA nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.