Constraint Vector Research Articles

On Combinatorial Properties of the Knapsack Problem

The knapsack problem with Boolean variables and a single constraint is studied. In the general case, this problem is NP-hard; for this reason, its exact solution requires the use of various search algorithms with the decomposition of the set of feasible solutions and computation of estimates of the objective function. Combinatorial formulas for computing and estimating the value of the objective function in various cases depending on the set of given parameters of the problem are derived. The case when the coefficients of the constraint vector coincide with the coefficients of the objective function is considered. The relationship between the set of solutions of the problem and threshold functions of a certain type is revealed. The coefficients of the objective function, the coefficients of the constraint vector, and the knapsack size are used as parameters. The classical method of generating functions is used as the basic technique. The results obtained in this paper can be used, in particular, for estimating the complexity of search and decomposition methods of solving the problem and for developing such methods as auxiliary procedures.

WEDGING OF FRICTIONAL ELASTIC SYSTEMS

We consider discrete two-dimensional elastic systems with Coulomb friction contacts, and investigate the conditions that must be satisfied if these are to be capable of becoming ‘wedged’ --- i.e. of remaining with non-zero elastic deformations when all external loads have been removed. The condition for wedging is reduced to the requirement that a prescribed set of constraint vectors should fail to positively span the N-dimensional vector space of nodal displacements. We also show that the range of admissible wedged states increases monotonically with the coefficient of friction f and that there exists a unique critical coefficient fw such that wedging is impossible for f < fw and possible for f > fw.

Penalty Function Method and Regularization in the Analysis of Improper Convex Programs

We consider the questions of correction of improper convex programs, first of all, problems with inconsistent systems of constraints. Such problems often arise in the practice of mathematical simulation of specific applied settings in operations research. Since improper problems are rather frequent, it is important to develop methods of their correction, i.e., methods of construction of solvable models that are close to the original problems in a certain sense. Solutions of these models are taken as generalized (approximation) solutions of the original problems. We construct the correcting problems using a variation of the right-hand sides of the constraints with respect to the minimum of a certain penalty function, which, in particular, can be taken as some norm of the vector of constraints. As a result, we obtain optimal correction methods that are modifications of the (Tikhonov) regularized method of penalty functions. Special attention is paid to the application of the exact penalty method. Convergence conditions are formulated for the proposed methods and convergence rates are established.

An optimized analytical solution for geostationary debris removal using solar sails

Debris in geostationary Earth orbits is often uncontrollable, consisting of launch-vehicle upper stages and non-operational payloads. To help mitigate orbital debris congestion, a solar-sailing satellite concept is proposed which retrieves and relocates large debris for placement into the “graveyard” orbit above the geostationary regime. This work derives an analytical deorbit solution based on Lyapunov control theory combined with the calculus of variations. A dynamic constraint vector is introduced as a result, which dictates the orbital response of a spacecraft to some controlled perturbation. The resulting controller is simulated for various solar sailing platforms to characterize deorbit capability based on a system's area to mass ratio. User design parameters in these solutions are then optimized using a Particle Swarm Optimizer (PSO) to produce robust, locally time optimal solutions for orbital debris removal using solar sails. Solar sail deorbit times are shown to decrease as the reflective area increases, with additional performance dependencies based on the Earth's relative position from the sun.

Robust optimization of uncertain structures based on interval closeness coefficients and the 3D violation vectors of interval constraints

In this paper, a novel direct interval robust optimization approach is proposed so that the robust optimal design vectors for structures with interval uncertainties can be achieved. A new concept of interval closeness coefficient is proposed to describe the relative positional relationship between the boundaries of two intervals. Subsequently, the explicit formulae for calculating the four interval closeness coefficients between the boundaries of an interval constraint mechanical performance index and those of its corresponding given interval constant are put forward. Consequently, the 3D violation vectors of different interval constraints can be calculated, and the feasibility of a design vector can be evaluated by its total 3D violation vector of all interval constraints. Finally, various design vectors are directly ranked according to the preferential guidelines considering the robustness of all the mechanical performance indices of uncertain structures, which is realized by integrating the Kriging technique and nested genetic algorithm. Unlike the traditional robust optimization of structures involving interval uncertainties, the proposed method can avoid the complicated model transformation process and ensure the robustness of all the mechanical performance indices of the optimal solution. Two examples are thoroughly investigated, the results of which demonstrate the applicability and advantages of the proposed approach.

Adaptive Syncretic Attention for Constrained Image Captioning

Recently, deep learning approaches for image captioning have gained a lot of attention and achieved overwhelming progress. In this paper, we propose a novel model which simultaneously explores a better representation of images and the relationship between visual and semantic information. The model consists of three parts: an Adaptive Syncretic Attention (ASA) mechanism, a LSTM + MLP mimic constraint network and a multimodal layer. In the ASA, we integrate local semantics features captured by region proposal network with time-varying global visual features through attention mechanism. In the LSTM + MLP mimic constraint network, we designed a network which consists of Multilayer Perceptron (MLP) and Long Short Term Memory (LSTM) model. During test process, this network generates a Mimic Constraint Vector for each test image. Further, we combine textual and visual information in our multimodal layer. Based on these three parts, our full model is capable of both capturing meaningful local features and generating sentence that is more relevant to image content. We evaluate our model on two popular datasets (i.e., Flickr30k and MSCOCO datasets). The results show that each module can improve our model. Moreover, our entire model is on par with or even better than the state-of-the-art methods.

Electric sail trajectory design with Bezier curve-based shaping approach

The aim of this paper is to propose a shape-based method in which the concept of Bezier curve is used to efficiently design the three-dimensional interplanetary trajectory of a spacecraft whose primary propulsion system is an Electric Solar Wind Sail. The latter is a propellantless propulsion concept that consists of a spinning grid of tethers, kept at a high positive potential by a power source and maintained stretched by the centrifugal force. The proposed approach approximates the time variation of the components of the spacecraft position vector using a Bezier curve function, whose geometric coefficients are calculated by optimizing the total flight time with standard numerical methods and enforcing the boundary conditions of a typical interplanetary rendezvous mission. The paper also discusses a geometrical approach to include, in the optimization process, the propulsive acceleration vector constraints obtained with the latest Electric Solar Wind Sail thrust model.

光场相机精确色彩矢量约束下的超分辨率算法

光场相机精确色彩矢量约束下的超分辨率算法

Dynamic model for internally carried air-launched rocket

For rigid-flexible coupling multi-body with variable topology, such as the system of internally carried air-launched or heavy cargo airdrop, in order to construct a dynamic model with unified form, avoid redundancy in the modeling process and make the solution independent, a method based on the equivalent rigidization model was proposed. It divides a system into independent subsystems by cutting off the joints, of which types are changed with the operation process of the system. And models of different subsystems can be constructed via selecting suitable modeling methods. Subsystem models with flexible bodies are on the basis of the equivalent rigidization model which replaces the flexible bodies with the virtual rigid bodies. And the solution for sanction, which is based on the constraints force algorithm (CFA) and vector mechanics, can be independent on the state equations. The internally carried air-launched system was taken as an example for verifying validity and feasibility of the method and theory. The dynamic model of aircraft-rocket-parachute system in the entire phase was constructed. Comparing the modeling method with the others, the modeling process was programmed; and form of the model is unified and simple. The model, method and theory can be used to analyze other similar systems such as heavy cargo airdrop system and capsule parachute recovery system.

Compensation for aircraft effects of magnetic tensor gradiometer with nonlinear least square method

Abstract Based on the detailed analysis of magnetic interference fields, mathematical model of compensation coefficients is constructed and nonlinear compensation method is proposed for magnetic tensor gradiometer in this paper. Firstly, a mathematical model containing compensation coefficients of magnetic tensor gradiometer is deduced by extending the Leliak method from the magnetic interference vector field compensation model. Secondly, four different types of constraint conditions, which are independent of the attitude of the magnetic tensor gradiometer and invariant with maneuvers, and a class of vector constraints are used to establish proper vector and tensor constraint equations. Then the compensation coefficients are solved using the nonlinear least square algorithm. Effectiveness and convergence of the proposed method are proved by simulation experiment, and the magnetic interference signals of aircraft effects are decreased remarkably.

Adaptive conjugate single-loop method for efficient reliability-based design and topology optimization

The single-loop approach (SLA) for reliability-based design optimization (RBDO) is one of the most efficient schemes for optimization problems with linear and weak nonlinear probabilistic constraints. However, it may produce unstable results or increase computational efforts when using the ordinary search direction to determine the optimum design in RBDO problems with highly nonlinear probabilistic constraints. The conjugate gradient (CG) is a promising sensitivity vector for locating the most probable point (MPP) of highly nonlinear concave performance functions. However, the MPP computation using the CG may require a high computational burden for convex constraints To overcome the drawbacks of the SLA the adaptive conjugate single-loop approach (AC-SLA) is proposed for RBDO problems with a large variety of nonlinear constraints. The sensitivity vector of the probabilistic constraints is adaptively computed using the CG vector with a dynamical conjugate scalar factor (DCF). The DCF is adjusted within the range from 0 to 2 using two adaptive coefficients, which are adapted based on the new and previous points. Moreover, the Lyapunov exponents are developed as a general tool for detecting the robustness of different MPP approximation algorithms. The method is also applied to solve a reliability-based topology optimization (RBTO) problem. The ability of the AC-SLA in six RBDO benchmark problems, one applicable to an RBDO aircraft engineering problem and one for RBTO problem, is compared in terms of both robustness and efficiency using the SLA, performance measure approach (PMA), reliability index approach (RIA), and sequential optimization and reliability assessment (SORA). The numerical experiments demonstrate that the convergence capability of the proposed AC-SLA is significantly superior to the PMA, RIA, and SORA. The computational effort of the AC-SLA is significantly reduced, with stable results.

Hamiltonian analysis of 4-dimensional spacetime in Bondi-like coordinates**Supported by the National Natural Science Foundation of China (11690022)

We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,xa,a = 2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T±(2), whose Lie algebra (1,3) is decomposed into and t±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in BF theory. General relativity can be reformulated as the four-dimensional coframe () and connection () dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e−, e+ and two spacelike 1-forms e2, e3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about . The consistency conditions of the primary constraints can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints and 40 second-class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n0, l0, and are Ricci identities. The equations of motion of the canonical variables have also been shown.

Disclinations without gradients: A nonlocal model for topological defects in liquid crystals

Nematic liquid crystals composed of rod-like molecules have an orientational elasticity that accounts for the energetics of the molecular orientation. This elasticity can be described by a unit vector field; the unit vector constraint interacts with even fairly simple boundary conditions to cause disclination defects. Disclinations are entirely a topological consequence of the kinematic constraint, and occur irrespective of the particular energetic model. Because disclinations are topological defects, they cannot be regularized by adding higher gradients, as in phase-field models of interface defects. On the contrary, the higher gradient terms would cause even greater singularities in the energy. In this paper, we formulate an integral-based nonlocal regularized energy for nematic liquid crystals. Our model penalizes disclination cores and thereby enforces a finite width, while the integral regularization ensures that the defect core energy is bounded and finite. The regularization at the same time tends to the standard gradient-based energies away from the disclination, as well as building in the head-tail symmetry. We characterize the formulation in its ability to describe disclinations of various strengths, and then apply it to examine: (1) the stability and decomposition of various disclinations, and the competition between bend and splay energies in determining the relative stability of integer and half-integer disclinations (2) the coalescence of a +12 and−12 disclination pair; we find the disclinations do not move at the same velocities towards each other, suggesting that the asymmetry of the director field plays a dominant role despite the equal-and-opposite topological strengths of the disclinations.

Generating Functions in the Knapsack Problem

The knapsack problem with Boolean variables and a single constraint is considered. Combinatorial formulas for calculating and estimating the cardinality of the set of feasible solutions and the values of the functional in various cases depending on given parameters of the problem are derived. The coefficients of the objective function and of the constraint vector and the knapsack size are used as parameters. The baseline technique is the classical method of generating functions. The results obtained can be used to estimate the complexity of enumeration and decomposition methods for solving the problem and can also be used as auxiliary procedures in developing such methods.

Polynomial BF -type action for general relativity and anti-self-dual gravity

We report a gravitational $BF$-type action principle propagating two (complex) degrees of freedom that, besides the gauge connection and the $B$ field, only employs an additional Lagrange multiplier. The action depends on two parameters and remarkably is polynomial in the $B$ field. For a particular choice of the involved parameters the action provides an alternative description of (complex) general relativity with a nonvanishing cosmological constant, whereas another choice corresponds to anti-self-dual gravity. Generic values of the parameters produce "close neighbors" of general relativity, although there is a peculiar choice of the parameters that leads to a Hamiltonian theory with two scalar constraints. Given the nontrivial form of the resulting scalar constraint for these models, we consider a more general setting where the scalar constraint is replaced with an arbitrary analytic function of some fundamental variables and show that the Poisson algebra involving this constraint together with the Gauss and vector constraints of the Ashtekar formalism closes, thus generating an infinite family of gravitational models that propagate the same number of degrees of freedom as general relativity.

Stochastic C-GNet Environment Modeling and Path Planning Optimization in a Narrow and Long Space

This study proposes a novel method of optimal path planning in stochastic constraint network scenarios. We present a dynamic stochastic grid network model containing semienclosed narrow and long constraint information according to the unstructured environment of an underground or mine tunnel. This novel environment modeling (stochastic constraint grid network) computes the most likely global path in terms of a defined minimum traffic cost for a roadheader in such unstructured environments. Designing high-dimensional constraint vector and traffic cost in nodes and arcs based on two- and three-dimensional terrain elevation data in a grid network, this study considers the walking and space constraints of a roadheader to construct the network topology for the traffic cost value weights. The improved algorithm of variation self-adapting particle swarm optimization is proposed to optimize the regional path. The experimental results both in the simulation and in the actual test model settings illustrate the performance of the described approach, where a hybrid, centralized-distributed modeling method with path planning capabilities is used.

Probability-one homotopy methods for constrained clustering

Abstract Many algorithms for constrained clustering have been developed in the literature that aim to balance vector quantization requirements of cluster prototypes against the discrete satisfaction requirements of constraint (must-link or must-not-link) sets. A significant amount of research has been devoted to designing new algorithms for constrained clustering and understanding when constraints help clustering. However, no method exists to systematically characterize solution sets as constraints are gently introduced and how to assist practitioners in choosing a sweet spot between vector quantization and constraint satisfaction. A homotopy method is presented that can smoothly track solutions from unconstrained to constrained formulations of clustering. Beginning the homotopy zero curve tracking where the solution is (fairly) well-understood, the curve can then be tracked into regions where there is only a qualitative understanding of the solution set, finding multiple local solutions along the way. Experiments demonstrate how the new homotopy method helps identify better tradeoffs and reveals insight into the structure of solution sets not obtainable using pointwise exploration of parameters.

Dual-polarized GPS antenna array algorithm to adaptively mitigate a large number of interference signals

Intentional or unintentional interference experienced by the Global Positioning System (GPS) is a significant concern for GPS-based critical infrastructures and services including aviation. An effective way to mitigate GPS interference is to utilize a GPS antenna array capable of electronically changing its gain pattern. Although a conventional GPS antenna array consists of single-polarized antenna elements, a dual-polarized antenna array has the potential to mitigate approximately twice the number of interference signals in the spatial domain as a single-polarized array because of the additional degrees of freedom provided. This paper proposes an adaptive beamforming algorithm using a dual-polarized GPS antenna array for mitigation of interference signals with various polarizations. In this paper, a dual-polarized antenna element specifically refers to two co-located crossed linearly polarized dipole antennas. The proposed minimum-variance-distortionless-response (MVDR)-based space–time adaptive processing (STAP) method utilizes a novel constraint vector that is specially designed for a dual-polarized GPS array. As the proposed constraint vector considers realistic radiation patterns of the antenna, the performance of the proposed method in terms of the signal-to-interference-plus-noise power ratio (SINR) is noticeably superior to that of previous methods under a representative interference scenario. The performance of the proposed method is compared with three previous methods utilizing dual-polarized antenna arrays.

Postoptimization of the model of water supply for urban and industrial agglomeration

A case study of the tools used by an analyst of the economic aspects of the operation of the water supply network has been undertaken in this paper. All issues discussed here are formulated by using degenerated linear programming models ( PL ). Below, it is noted that the linear dependence of binding constraints ( CO ) distorts standard postoptimization procedures in PL. This observed fact makes postoptimization analysis mostly unhelpful for an average analyst due to problems with the int erpretation of ambiguous sensitivity reports which are obtained from popular computer packages. In standard postoptimization methods, changes to single parameters of the right-hand vector CO are analyzed or referred to parametric linear programming that unfortunately requires prior knowledge of mathematically and economically justified vectors of changes of right-hand sides CO. Therefore, it is suggested that modifications are introduced to some of the postoptimization procedures in this work. For issues in the field of hydrology, the following were presented: interpretation and methods of generating justified vectors of changes of right-hand sides of limiting conditions. And so, the procedure of generating infinitely many solutions of the dual issue based on certain vectors orthogonal to the vector of right-hand sides of constraint conditions was demonstrated. Furthermore, the same orthogonal vectors were used to obtain nodal solutions of the dua0l model and the corresponding vectors of changes of the entire right-hand sides of the constraint conditions. Then, managerial interpretation was applied to this way of proceeding. The methods presented in the work serve to improve the functioning of the system of water supply.

ALTERNANCE METHOD IN VECTOR PROBLEMS OF PARAMETRIC OPTIMIZATION OF SYSTEMS WITH DISTRIBUTED PARAMETERS

Constructive methods are proposed for solving a wide range of vector problems of optimal control of systems with distributed parameters (DPS) under the conditions when an accuracy of uniform approximation to the required final state of an object is prescribed on a set of spatial arguments of a controlled quantity. After the transition to the relative equivalent estimates of particular efficiency criteria the problems of vector (multi-criteria) optimization are reduced to a single-criterion version in the form of a variational problem with an integral quality criterion and with the new restrictions on the final values of the additional phase variables of the extended DPS’ model. Vector (combined) constraints on the final state of the DPS are replaced by one constraint on their linear combination, which is presented in the uniform metric on the extended set of arguments, including (besides the spatial variables) the weight factors of the summable components that vary within the standard simplex. The problems of vector (multichannel) control by one or by a system of interconnected distributed objects are considered under the conditions of a specific requirement of the same duration of the control process for all control actions. Further procedures of preliminary parameterization of control actions carried out using well-known analytical conditions of optimality provide accurate reduction of the initial statements to complicated modifications of semi-infinite programming problems, which are applied to computational algorithms developed earlier by the authors for the scalar version of the alternance method of searching for the desired extremals based on their Chebyshev properties. An example of independent solution of a vector problem of optimal control of an object of technological thermophysics is given.