System Of Linear Constraints Research Articles

New characterizations of Hoffman constants for systems of linear constraints

We give a characterization of the Hoffman constant of a system of linear constraints in $${{\mathbb {R}}}^n$$ relative to a reference polyhedron $$R\subseteq {{\mathbb {R}}}^n$$ . The reference polyhedron R represents constraints that are easy to satisfy such as box constraints. In the special case $$R = {{\mathbb {R}}}^n$$ , we obtain a novel characterization of the classical Hoffman constant. More precisely, suppose $$R\subseteq \mathbb {R}^n$$ is a reference polyhedron, $$A\in {{\mathbb {R}}}^{m\times n},$$ and $$A(R):=\{Ax: x\in R\}$$ . We characterize the sharpest constant $$H(A\vert R)$$ such that for all $$b \in A(R) + {{\mathbb {R}}}^m_+$$ and $$u\in R$$ $$\begin{aligned} {\mathrm {dist}}(u, P_{A}(b)\cap R) \le H(A\vert R) \cdot \Vert (Au-b)_+\Vert , \end{aligned}$$ where $$P_A(b) = \{x\in {{\mathbb {R}}}^n:Ax\le b\}$$ . Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.

Modified Stojkovic-Stanimirovic method to find redundant constrains in linear programming problems

Important techniques in linear programming are modeling and solving practical optimization. A redundant constraint is a constraint that can be remove from a system of linear constraints without changing the feasible region. Method for identifying redundant constraints which in the process only uses the objective function and inequality constraints, i.e., Stojkovic-Stanimirovic method. In this paper we give a modified method of Stojkovic-Stanimirovic method to detect redundant constraints in linear programming problems. The advantages of employing the propose method is still to use the different objective function on linear programming associated the same feasible set have the same result. To clarify these method is given an example cannot be solve using detect Stojkovic-Stanimirovic method but can be solve using propose method.

The Laurent Coefficients of the Hilbert Series of a Gorenstein Algebra

By a theorem of R. Stanley, a graded Cohen-Macaulay domain A is Gorenstein if and only if its Hilbert series satisfies the functional equation:where, d is the Krull dimension and a is the -invariant of A. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of at t = 1. The main idea consists of examining the graded algebra of formal power series in the variable x that fulfill the condition . As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of Gould and Carlitz on power sums of symmetric number triangles is established.

Search Directions in Infeasible Newton’s Method for Computing Weighted Analytic Center for Linear Matrix Inequalities

Four different search directions for Infeasible Newton’s method for computing the weighted analytic center defined by a system of linear matrix inequality constraints are studied. Newton’s method is applied to find the weighted analytic center and the starting point can be infeasible, that is, outside the feasible region determined by the linear matrix inequality constraints. More precisely, Newton’s method is used to solve system of equations given by the KKT optimality conditions for the weighted analytic center. The search directions for the Newton’s method considered are the ZY, ZY+YZ, Z−1 and NT methods that have been used in semidefinite programming. Backtracking line search is used for the Newton’s method. Numerical experiments are used to compare these search direction methods on randomly generated test problems by looking at the iterations and time taken to compute the weighted analytic center. The starting points are picked randomly outside the feasible region. Our numerical results indicate that ZY+YZ and ZY are the best methods. The ZY+YZ method took the least number of iterations on average while ZY took the least time on average and they handle weights better than the other methods when some of the weights are very large relative to the other weights. These are followed by NT method and then Z−1 method.

Algorithms for Solving the Systems of Linear Constraints with Integer Coefficients in the Set {0, 1}

Algorithms for Solving the Systems of Linear Constraints with Integer Coefficients in the Set {0, 1}

Quantum and non-signalling graph isomorphisms

We introduce the (G,H)-isomorphism game, a new two-player non-local game that classical players can win with certainty iff the graphs G and H are isomorphic. We then define quantum and non-signalling isomorphisms by considering perfect quantum and non-signalling strategies for this game. We prove that non-signalling isomorphism coincides with fractional isomorphism, giving the latter an operational interpretation. We show that quantum isomorphism is equivalent to the feasibility of two polynomial systems obtained by relaxing standard integer programs for graph isomorphism to Hermitian variables. Finally, we provide a reduction from linear binary constraint system games to isomorphism games. This reduction provides examples of quantum isomorphic graphs that are not isomorphic, implies that the tensor product and commuting operator frameworks result in different notions of quantum isomorphism, and proves that both relations are undecidable.

QuadriFlow: A Scalable and Robust Method for Quadrangulation

AbstractQuadriFlow is a scalable algorithm for generating quadrilateral surface meshes based on the Instant Field‐Aligned Meshes of Jakob et al. (ACM Trans. Graph. 34(6):189, 2015). We modify the original algorithm such that it efficiently produces meshes with many fewer singularities. Singularities in quadrilateral meshes cause problems for many applications, including parametrization and rendering with Catmull‐Clark subdivision surfaces. Singularities can rarely be entirely eliminated, but it is possible to keep their number small. Local optimization algorithms usually produce meshes with many singularities, whereas the best algorithms tend to require non‐local optimization, and therefore are slow. We propose an efficient method to minimize singularities by combining the Instant Meshes objective with a system of linear and quadratic constraints. These constraints are enforced by solving a global minimum‐cost network flow problem and local boolean satisfiability problems. We have verified the robustness and efficiency of our method on a subset of ShapeNet comprising 17,791 3D objects in the wild. Our evaluation shows that the quality of the quadrangulations generated by our method is as good as, if not better than, those from other methods, achieving about four times fewer singularities than Instant Meshes. Other algorithms that produce similarly few singularities are much slower; we take less than ten seconds to process each model. Our source code is publicly available.

Algorithm for the simplification of solution to discrete optimization problems

Typically, the search for solutions in discrete optimization problems is associated with fundamental computational difficulties. The known methods of accurate or approximated solution of such problems are studied talking into consideration their belonging to so-called problems from P and NP class (algorithms for polynomial and exponential implementation of solution). Modern combinatorial methods for practical solution of discrete optimization problems are focused on the development of algorithms which allow obtaining an approximated solution with guaranteed evaluation of deviations from the optimum. Simplification algorithms are an effective technique of the search for solutions to an optimization problem. If we make a projection of a multi-dimensional process onto a two-dimensional plane, this technique will make it possible to clearly display a set of all solutions to the problem in graphical form. The method for simplification of the combinatorial solution to the discrete optimization problem was proposed in the framework of this research. It is based on performance of decomposition of a system that reflects the system of constraints of the original five-dimensional original problem on a two-dimensional plane. This method enables obtaining a simple system of graphic solutions to a complex problem of linear discrete optimization. From the practical point of view, the proposed method enables us to simplify computational complexity of optimization problems of such a class. The applied aspect of the proposed approach is the use of obtained scientific results in order to provide a possibility to improve the typical technological processes, described by systems of linear equations with existence of systems of linear constraints. This is a prerequisite for subsequent development and improvement of similar systems. In this study, the technique for decomposition of a discrete optimization system through projection of an original problem on two-dimensional coordinate planes was proposed. In this case, the original problem is transformed to a combinatorial family of subsystems, which makes it possible to obtain a system of graphic solutions to a complex problem of linear discrete optimization.

Environmental graphic method for creating area of permissible impact

The methods of ecological optimization of influences developed by the author are presented, including: - graphic method of creation of area of admissible influences by the solution of problems of linear programming; - model of identification of dangerous influences simplex method; - method of formation of steady control systems of ecological safety of construction. On simple numerical examples possibilities of mathematical modeling of environmental pressures at stages of life cycle of construction objects are illustrated. In the paper, linear equations of action with constraints and with two variable factors of influence were considered. If we go over to linear relationships with three variable factors of influence, then they will describe the plane in the three-dimensional impact space; the system of linear constraints is a polyhedron, as the domain of permissible impacts in the three-dimensional impact space. Studies carried out by the graphical method of constructing the region of permissible impacts in solving ecological problems of linear programming have shown its effectiveness and visibility, in comparison with the results obtained by calculation. The most effective is the development of environmental safety management systems for construction related to sources, definitions and categories of wastes, as well as the composition and treatment of waste streams, in order to promote waste prevention and the establishment and implementation of mechanisms to prevent generation and minimize waste, as well as recovery and recycling systems waste construction industry.

Optimal controller design based on minimum principle

Optimal control theory is the foundation of the modern control theory, the minimum principle in optimal control theory has a very important position, using the minimum principle to design an adaptive controller, the controller integration advantages of the principle of minimum is not affected by the control system of linear or nonlinear constraints, and the end state and free time, is accused of quantity can be controlled and are free to wait for a characteristic, using the minimum controller application example and simulation, the results show that the minimum principle of the designed controller has the ideal control effect.

Improvement of field method and its application to integrating motion equation in Riemann-Cartan space

Like the Hamilton-Jacobi method, the Vujanović field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanović field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanović field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanović field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m (m n) first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution (the complete solution is a special case of m=n) of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanović field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanović field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subjected to an active force, its motion equation in its Riemann-Cartan configuration space can be obtained by a first order nonlinear nonholonomic mapping. Since the motion equations in Riemann-Cartan configuration space contain quasi-speeds, they are often considered to be difficult to solve directly. In this paper we give a briefing of how to construct the motion equations of a first order linear nonholonomic constraint system in its Riemann-Cartan configuration space, and how to obtain the first integrals of the motion equations in the Riemann-Cartan configuration space by the improved field method. This is an effective method to study some nonholonomic nonconservative motions.

Solution Stability of a Linearly Perturbed Constraint System and Applications

Linear complementarity problems and affine variational inequalities have been intensively investigated by different methods. Recently, some authors have shown that solution stability of these problems with respect to total perturbations can be effectively studied via a generalized linear constraint system. The present paper focuses on characterizing stability properties of the solution map of a linearly perturbed generalized linear constraint system. The obtained results lead to several stability conditions for parametric linear complementarity problems and affine variational inequalities in explicit forms.

Knapsack with variable weights satisfying linear constraints

We introduce a variant of the knapsack problem, in which the weights of items are also variables but must satisfy a system of linear constraints, and the capacity of knapsack is given and known. We discuss two models: (1) the value of each item is given; (2) the value-to-weight ratio of each item is given. The goal is to determine the weight of each item, and to find a subset of items such that the total weight is no more than the capacity and the total value is maximized. We provide several practical application scenarios that motivate our study, and then investigate the computational complexity and corresponding algorithms. In particular, we show that if the number of constraints is a fixed constant, then both problems can be solved in polynomial time. If the number of constraints is an input, then we show that the first problem is NP-Hard and cannot be approximated within any constant factor unless \(\mathrm{P}= \mathrm{NP}\), while the second problem is NP-Hard but admits a polynomial time approximation scheme. We further propose approximation algorithms for both problems, and extend the results to multiple knapsack cases with a fixed number of knapsacks and identical capacities.

New techniques for linear arithmetic: cubes and equalities

We present several new techniques for linear arithmetic constraint solving. They are all based on the linear cube transformation, a method presented here, which allows us to efficiently determine whether a system of linear arithmetic constraints contains a hypercube of a given edge length. Our first findings based on this transformation are two sound tests that find integer solutions for linear arithmetic constraints. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width. Experimental results confirm that our tests are superior on these instances compared to several state-of-the-art SMT solvers. We also discovered that the linear cube transformation can be used to investigate the equalities implied by a system of linear arithmetic constraints. For this purpose, we developed a method that computes a basis for all implied equalities, i.e., a finite representation of all equalities implied by the linear arithmetic constraints. The equality basis has several applications. For instance, it allows us to verify whether a system of linear arithmetic constraints implies a given equality. This is valuable in the context of Nelson–Oppen style combinations of theories.

An algorithm for constructing the skeleton graph of degenerate systems of linear inequalities.

Derive the quantitative predictions of constraint-based models require of conversion algorithms to enumerate and construct the skeleton graph conformed by the extreme points of the feasible region, where all constraints in the model are fulfilled. The conversion is problematic when the system of linear constraints is degenerate. This paper describes a conversion algorithm that combines the best of two methods: the incremental slicing of cones that defeats degeneracy and pivoting for a swift traversal of the set of extreme points. An extensive computational practice uncovers two complementary classes of conversion problems. The two classes are distinguished by a practical measure of complexity that involves the input and output sizes. Detailed characterizations of the complexity classes and the corresponding performances of the algorithm are presented. For the benefit of implementors, a simple example illustrates the stages of the exposition.

An algorithm for solving two-sided interval system of max-plus linear equations

We study six types of solutions (weak, strong, tolerance, control, Left-localized, and Right-localized solutions) to a two-sided interval system of max-plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions. These conditions are in the form of two-sided systems of max-plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems. Therefore, we could work on the interval inclusion linear systems, instead of directly solving these six types of solutions themselves. An optimization problem with the two-sided interval system of max-plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems.

Making schedules in multiprocessing systems with several additional resources

The problem of making a feasible schedule in a preemptive multiprocessing system with identical processors and several types of additional recourses is considered in the case when the task execution intervals are given and the durations of task execution linearly depend on the amount of the additional resource allocated to them. Algorithms based on reducing this problem to a network flow problem and a system of linear constraints are developed.

Vivace

The solution of large sparse systems of linear constraints is at the base of most interactive solvers for physically-based animation of soft body dynamics. We focus on applications with hard and tight per-frame resource budgets, such as video games, where the solution of soft body dynamics needs to be computed in a few milliseconds. Linear iterative methods are preferred in these cases since they provide approximate solutions within a given error tolerance and in a short amount of time. We present a parallel randomized Gauss-Seidel method which can be effectively employed to enable the animation of 3D soft objects discretized as large and irregular triangular or tetrahedral meshes. At the beginning of each frame, we partition the set of equations governing the system using a randomized graph coloring algorithm. The unknowns in the equations belonging to the same partition are independent of each other. Then, all the equations belonging to the same partition are solved at the same time in parallel. Our algorithm runs completely on the GPU and can support changes in the constraints topology. We tested our method as a solver for soft body dynamics within the Projective Dynamics and Position Based Dynamics frameworks. We show how the algorithmic simplicity of this iterative strategy enables great numerical stability and fast convergence speed, which are essential features for physically based animations with fixed and small hard time budgets. Compared to the state of the art, we found our method to be faster and scale better while providing stabler solutions for very small time budgets.

Constrained fits with non-Gaussian distributions

Non-normally distributed data are ubiquitous in many areas of science, including high-energy physics. We present a general formalism for constrained fits, also called data reconciliation, with data that are not normally distributed. It is based on Bayesian reasoning and implemented via MCMC sampling. We show how systems of both linear and non-linear constraints can be efficiently treated. We also show how the fit can be made robust against outlying observations. The method is demonstrated on a couple of examples ranging from material flow analysis to the combination of non-normal measurements. Finally, we discuss possible applications in the field of event reconstruction, such as vertex fitting and kinematic fitting with non-normal track errors.

Conditionally exponential random models for individual properties and network structures: Method and application

Exponential random models have been widely adopted as a general probabilistic framework for complex networks and recently extended to embrace broader statistical settings such as dynamic networks, valued networks or two-mode networks. Our aim is to provide a further step into the generalization of this class of models by considering sample spaces which involve both families of networks and nodal properties verifying combinatorial constraints. We propose a class of probabilistic models for the joint distribution of nodal properties (demographic and behavioral characteristics) and network structures (friendship and professional partnership). It results in a general and flexible modeling framework to account for homophily in social structures. We present a Bayesian estimation method based on the full characterization of their sample spaces by systems of linear constraints. This provides an exact simulation scheme to sample from the likelihood, based on linear programming techniques. After a detailed analysis of the proposed statistical methodology, we illustrate our approach with an empirical analysis of co-authorship of journal articles in the field of neuroscience between 2009 and 2013.