System Of Linear Inequalities Research Articles

Представление торговых сигналов на основе адаптивной скользящей средней Кауфмана в виде системы линейных неравенств

This paper considers the adaptation of problem of strong separability for decisions about buying or selling financial assets, such as equities, currencies, futures, etc. on the stock exchange. There were constructed two systems of linear inequalities that define the regions in n-dimensional space. These systems describe the expert trading signals that based on adaptive moving average of Kaufman.

Iterative solution of inconsistent systems of linear inequalities

AbstractIn 1980 S.P. Han proposed a finitely terminating (in exact arithmetic) algorithm for solving an inconsistent system of linear inequalities in a least squares sense. This algorithm uses a singular value decomposition of a submatrix of the problem matrix on each iteration, making it impractical for all but only smaller problems. In this paper we show that a modification of Han's algorithm allows us to introduce an iterative approximation to the singular value decomposition solution. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational relation problems: existence of solutions and fixed points of contraction mappings

We propose a new approach to study variational relation problems. Namely, we apply Mizoguchi and Takahashi’s fixed point theorem of contraction mappings and an error bound of a system of linear inequalities to establish existence conditions for a variational relation problem in which the variational relation linearly depends on the decision variable. Then we develop an algorithm to compute a solution of a linear variational relation problem. MSC:49J52, 47H10.

Interpretation of contradictory images by means of systems of linear inequalities

We consider the problem of interpretation of three-dimensional images from their flat projections up to the set of visible faces. For projections of convex polyhedra, we present an interpretation algorithm based on maximal feasible subsystems of a certain infeasible system of linear inequalities modeling the visibility requirement for faces. A number of model examples are given; in particular, the algorithm is applied to the interpretation of the Necker cube.

On r-Edge-Connected r-Regular Bricks and Braces and Inscribability

A classical result due to Steinitz states that a graph is isomorphic to the graph of some 3-dimensional polytope P if and only if it is planar and 3-connected. If a graph G is isomorphic to the graph of a 3-dimensional polytope inscribed in a sphere, it is said to be of inscribable type. The problem of determining which graphs are of inscribable type dates back to 1832 and was open until Rivin proved a characterization in terms of the existence of a strictly feasible solution to a system of linear equations and inequalities which we call sys(G), which, surprisingly, also appears in the context of the Traveling Salesman Problem. Using such a characterization, various classes of graphs of inscribable type can be described. Dillencourt and Smith gave a characterization of 3-connected 3-regular planar graphs that are of inscribable and a linear-time algorithm for recognizing such graphs. In this paper, their results are generalized to r-edge-connected r-regular graphs for odd r ≥ 3 in the context of the existence of strictly feasible solutions to sys(G). An answer to an open question raised by D. Eppstein concerning the inscribability of 4-regular graphs is also given.

A MIP-based timetable rescheduling formulation and algorithm minimizing further inconvenience to passengers

This paper presents a timetable rescheduling algorithm based on Mixed Integer Programming (MIP) formulation when train traffic is disrupted. We minimize further inconvenience to passengers instead of consecutive delays caused by the disruption, since loss of time and satisfaction of the passengers are considered implicitly and insufficiently in the latter optimization. We presume that inconvenience of traveling by train consists of the traveling time on board, the waiting time at platforms and the number of transfers. Hence, the objective function is calculated on the positive difference between the inconvenience which each passenger suffers on his/her route in a rescheduled timetable and that in a planned timetable. The inconvenience-minimized rescheduling is often achieved at the cost of further train delays. Some trains dwell longer at a station to wait for extra passengers to come or to keep a connection, for instance. In the MIP model, train operation, each passenger’s behavior and the amount inconvenience are simultaneously expressed by a system of integer linear inequalities. As countermeasures against the disruption, changes of train types and rolling stock operation schedules at termini as well as changes of departing order of trains and assignment of a track to trains in stations are performed. We also consider capacities of a line between adjacent stations as well as those of a track in stations. We have conducted numerical experiments using actual data and have obtained better rescheduled timetables in terms of customer satisfaction within practical time in proper solution space.

Generation Investment Equilibria With Strategic Producers—Part I: Formulation

The first of this two-paper series proposes a methodology to characterize generation investment equilibria in a pool-based network-constrained electricity market, where the producers behave strategically. To this end, the investment problem of each strategic producer is represented using a bilevel model, whose upper-level problem determines the optimal investment and the supply offering curves to maximize its profit, and whose several lower-level problems represent different market clearing scenarios. This model is transformed into a mathematical program with equilibrium constraint (MPEC) through replacing the lower-level problems by their optimality conditions. The joint consideration of all producer MPECs, one per producer, constitutes an equilibrium problem with equilibrium constraints (EPEC). To identify the solutions of this EPEC, each MPEC problem is replaced by its Karush-Kuhn-Tucker (KKT) conditions, which are in turn linearized. The resulting mixed-integer linear system of equalities and inequalities allows determining the EPEC equilibria through an auxiliary MILP problem.

Signal Processing with Regularized Multistep Support Vector Method

A new method is proposed for processing the signal distorted by random noise. The processing model is based on a statistical regularization method, and the obtained system of linear equations and inequalities is solved using a multistep support vector method. An advantage of this approach is that the iterative nature of the algorithm makes it possible to take into account the a priori information on the solution represented by the inequalities. The results of numerical experiments showing the efficiency of the algorithm are given.

On Metric Regularity and the Boundary of the Feasible Set in Linear Optimization

This paper deals with semi-infinite linear inequality systems in ℝ n and studies the stability of the boundary of their feasible sets. We analyze the equivalence between the metric regularity of the inverse of the boundary set mapping, \(\mathcal{N}\), and the stability of the feasible set mapping in the sense of the maintenance of the consistency. In doing this we provide operational formulae for distances from points to some useful sets. We also include relationships between the regularity moduli corresponding to the mappings \(\mathcal{N}\) and the inverse, \(\mathcal{M}\), of the feasible set mapping, and prove their equality for finite systems and some special cases in the semi-infinite framework. Moreover, we provide conditions to assure that the metric regularity of \(\mathcal{N}\) is equivalent to the lower semi-continuity of the boundary set mapping, which is important because the latter property has many characterizations. Since the boundary of a feasible set may not be convex, we cannot make use of the general theory for mappings with convex graph, as for example, the Robinson–Ursescu theorem.

Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations

This paper examines the application of optimization and control ideas to the formulation of feature-preserving numerical methods, with particular emphasis on the conservative and bound-preserving remap (constrained interpolation) and transport (advection) of a single scalar quantity. We present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization-based remap (OBR) of mass density. The latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity, subject to a system of linear inequality constraints. The approximation of an exact mass update operator defines the target quantity, which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds. The latter are enforced by the system of linear inequalities. In so doing, the generic OBR formulation separates accuracy considerations from the enforcement of physical properties.We proceed to show how the generic OBR formulation yields the recently introduced flux-variable flux-target (FVFT) [1] and mass-variable mass-target (MVMT) [2] formulations of remap and then follow with a formal examination of their relationship. Using an intermediate flux-variable mass-target (FVMT) formulation we show the equivalence of FVFT and MVMT optimal solutions.To underscore the scope and the versatility of the generic OBR formulation we introduce the notion of adaptable targets, i.e., target quantities that reflect local solution properties, extend FVFT and MVMT to remap on the sphere, and use OBR to formulate adaptable, conservative and bound-preserving optimization-based transport algorithms for Cartesian and latitude/longitude coordinate systems. A selection of representative numerical examples on two-dimensional grids demonstrates the computational properties of our approach.

General linear formulations of stochastic dominance criteria

We develop and implement linear formulations of general Nth order stochastic dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean–variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.

Equilibrium computation of the Hart and Mas-Colell bargaining model

The 8-th problem raised by [Hart, S., Mas-Colell, A., 2010. Bargaining and cooperation in strategic form games. Journal of the European Economics Association 8 (1), 7–33], is solved. To be specific, I show that the set of SP equilibria can be determined by a finite number of systems of linear inequalities, which are efficiently solvable when there are two players. This is more or less surprising because the Hart and Mas-Colell bargaining model and the SP equilibrium both seem to be rather complicated, and it is well known that an arbitrary Nash equilibrium is hard to compute, even when there are only two players. Using this algorithm, it is shown that players of Prisoners’ Dilemma can cooperate to some extent in the Hart and Mas-Colell bargaining, and full cooperation is attainable as ρ, a parameter of this model, approaches to 1. Quantitative efficiency, i.e. price of anarchy, is also analyzed.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Equilibrium Computation of the Hart and Mas-Colell Bargaining Model

The 8th problem raised by [Hart and Mas-Colell (2010) Bargaining and Cooperation in Strategic Form Games. Journal of the European Economics Association, 8(1): 7-33], is solved. To be specific, I show that the set of SP equilibria can be determined by a finite number of systems of linear inequalities, which are efficiently solvable when there are two players. This is more or less surprising because the Hart and Mas-Colell bargaining model and the SP equilibrium both seem to be rather complicated, and it is well known that an arbitrary Nash equilibrium is hard to compute, even when there are only two players. Using this algorithm, it is shown that players of Prisoners' Dilemma can cooperate to some extent in the Hart and Mas-Colell bargaining, and full cooperation is attainable as \rho, a parameter of this model, approaches to 1. Quantitative efficiency, i.e. price of anarchy, is also analyzed.

Weak and strong solvability of interval linear systems of equations and inequalities

We consider weak and strong solvability of general interval linear systems consisting of mixed equations and inequalities with mixed free and sign-restricted variables. We generalize the well-known weak solvability characterizations by Oettli–Prager (for equations) and Gerlach (for inequalities) to a unified framework. In the same manner, we extend strong solvability theorems to general interval linear systems. Next, we propose a sufficient condition for checking strong solvability. We give an application in linear programming with interval data. By means of weak and strong solvability we determine limits of the optimal values for any form of the problem setting.

On quantified linear implications

A Quantified Linear Implication (QLI) is an inclusion query over two polyhedral sets, with a quantifier string that specifies which variables are existentially quantified and which are universally quantified. Equivalently, it can be viewed as a quantified implication of two systems of linear inequalities. In this paper, we provide a 2-person game semantics for the QLI problem, which allows us to explore the computational complexities of several of its classes. More specifically, we prove that the decision problem for QLIs with an arbitrary number of quantifier alternations is PSPACE-hard. Furthermore, we explore the computational complexities of several classes of 0, 1, and 2-quantifier alternation QLIs. We observed that some classes are decidable in polynomial time, some are NP-complete, some are coNP-hard and some are $\boldsymbol{\Pi}_{\textbf 2}^{\textbf P}$ -hard. We also establish the hardness of QLIs with 2 or more quantifier alternations with respect to the first quantifier in the quantifier string and the number of quantifier alternations. All the proofs that we provide for polynomially solvable problems are constructive, i.e., polynomial-time decision algorithms are devised that utilize well-known procedures. QLIs can be utilized as powerful modelling tools for real-life applications. Such applications include reactive systems, real-time schedulers, and static program analyzers.

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski-Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and we present the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.

103 単純トラス構造のグラフ的設計モデルに関する研究(OS1-1 デジタル設計・評価・製造システムI)

In this paper, we consider a discrete design problem of two-dimensional truss structures. Given a set of potential nodes and a set of potential bars connecting these nodes, i.e., a ground structure, we only treat solutions which are combinations of accepted nodes and accepted bars in the ground structure. We present a linear inequality system with integrality constraints to extract a subgraph from the ground structure, which tries to model the subgraph on an extension of the simple truss. The linear inequality system delivers a MIP (Mixed Integer Programming) formulation. We also conduct numerical experiments to examine the solutions obtained by solving the MIP problem, and report the results.

Linear inequalities via least squares

The Gaussian elimination method is usually used in solving problems related to systems of linear inequalities. The present review paper describes the application of the least-squares method to studying problems connected with linear inequalities (like redundant inequalities, theorems of alternative, mathematical programming). The minimum norm solution to the system of linear inequalities is found by solving a non-negative least-squares (NNLS) problem. A linear programming (LP) problem is transformed to the system of inequalities in several ways. By solving the corresponding NNLS problem an initial solution to the LP problem is found. The main ideas are explained by simple examples.

An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality [Formula: see text] is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.