Rectangular Constraint Research Articles

A study of interactive furnishing design in the context of digitalization with a view to functional requirements

Abstract Firstly, an optimization algorithm based on planar design constraints is proposed, which can automatically generate interior space furnishings. The main idea of the algorithm is to represent the room units in interior space furnishings with multiple rectangular combinations of polygons and to translate the constraints and rules in furnishing design into relevant rectangular constraints. In order to improve the applicability of the method, this study also proposes a hierarchical algorithm flow based on the above method. Then, the interactive toilet aid design and the new intelligent wheelchair design, and the intelligent and ecological interior furnishing design are used as examples to explore the practice of interactive furnishing design based on multi-functional needs, which meet the emotional, psychological, and physiological needs of users. Based on the data analysis of furnishing design by hierarchical algorithm, the results show that the average score of ordinary users in the comparison results with Humans is 13.37 the average score of designers is 11.72; the MIQP model can be used for interior space furnishing design of residential buildings and can be extended to apply to space furnishing of large-scale scenes, MIQP method is much higher than existing methods in terms of efficiency The MIQP method is much more efficient than existing methods. This study improves the quality and level of furnishing design and promotes innovation in the design industry.

Constraint relationship of ecosystem services in the Yangtze River Economic Belt, China.

Understanding the relationship among ecosystem services (ESs) is essential to promote ESs management and sustainable development. The relationship between ESs is mutual and can be expressed in terms of trade-offs, synergy, and constraints. The paper selected the InVEST model to assess the water yield (WY), soil conservation (SC), food production (FP), net primary productivity (NPP), and habitat quality (HQ) of the Yangtze River Economic Belt (YREB) and used the constraint line method to analyze the relationship of paired ecological services at three scales: landscape, watershed, and land category. The following conclusions were drawn: (1) during the study period, the spatial changes of the five ecological services in the YREB did not change much, but the spatial distribution of the ecological services was different. (2) From 2000 to 2015, the constraint line of YREB paired ecological services had a high degree of fit. Under the three levels of landscape, watershed, and land category, the YREB has a variety of constraint types, including negative lines, logarithms, paraboloids, humped shapes, and rectangles. (3) At the three levels, the constraint lines between FP, NPP, WY, and SC and HQ were stable rectangular constraints; WY-SC was hump shaped, FP-NPP, FP-SC, FP-WY, NPP-WY, and NPP-SC changed with the scale, showing different spatial scale changes. (4) The paired ESs directly determined the ecological constraint curve but under the combined effect of other factors, which would affect or change the constraint line. We discussed the effects of weather, topography, and economy on the constraint relationship, and found that all have different degrees of influence.

Guaranteed feasible control allocation using model predictive control

This paper proposes a guaranteed feasible control allocation method based on the model predictive control. Feasible region is considered to guarantee the determination of the desired virtual control signal using the pseudo inverse methodology and is described as a set of constraints of an MPC problem. With linear models and the given constraints, feasible region defines a convex polyhedral in the virtual control space. In order to reduce the computational time, the polyhedral can be approximated by a few axis aligned hypercubes. Employing the MPC with rectangular constraints substantially reduces the computational complexity. In two dimensions, the feasible region can be approximated by a few rectangles of the maximum area using numerical geometry techniques which are considered as the constraints of the MPC problem. Also, an active MPC is defined as the controller to minimize the cost function in the control horizon. Finally, several simulation examples are employed to illustrate the effectiveness of the proposed techniques.

Storage capacity in symmetric binary perceptrons

We study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric. We call these variants the rectangle-binary-perceptron (RPB) and the u-function-binary-perceptron (UBP). We show that, unlike for the usual step-function-binary-perceptron, the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space (for all rectangular constraints and for narrow enough u-function constraints, K < K*). We prove this fact (under two natural assumptions) using the first and second moment methods. We further use the second moment method to conjecture that solutions of the symmetric binary perceptrons are organized in a so-called frozen-1RSB structure, without using the replica method. We then use the replica method to estimate the capacity threshold for the UBP case when the u-function is wide K > K*. We conclude that full-step-replica-symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case.

Stochastic nonlinear resource allocation problem

In this paper, we deal with a resource allocation problem modeled as special case of 0-1 Quadratic Programs with joint probabilistic rectangular constraints (QPJPC) with normally distributed coefficients and independent matrix vector rows. We reformulate this problem as a completely positive problem. In addition, the optimal value of the latter problem converges to the optimal value of the original problem under certain conditions. Numerical experiments on randomly generated data are given.

Divide and Conquer treemaps: Visualizing large trees with various shapes

Most existing treemaps achieve the space utilization of a single geometrical area, mostly rectangle. Limiting visualization to rectangles could block the human capability on graph recognition, including orientation, shape and differentiation etc. To relax rectangular constraint, we propose a flexible enclosure approach with three algorithms. It partitions large hierarchical structures within a confined display area with different shapes for real-time applications. Our approach is based on the combination of Divide-and-Conquer method and the treemap paradigm. The partitioning algorithms generate three types of layouts with polygonal, angular and rectangular titling, which are flexible to be used separately or combined. We present technical details including the visualization results in the experiments and in the cases studies with real data sets. We evaluated the visualization based on graph drawing aesthetics and optimization criteria. Our usability study shows that (1) treemaps with layout variability support utilization of human capability in graph perception and (2) treemaps adopted in different shaped containers could have a positive impact on user satisfaction and awareness during visual data exploration.

A column generation approach for evaluating delivery efficiencies of collimator technologies in IMRT treatment planning

Collimator systems used in Intensity Modulated Radiation Therapy can form different geometric aperture shapes depending on their physical capabilities. We compare the efficiency of using regular, rotating and dual multileaf collimator (MLC) systems under different combinations of consecutiveness, interdigitation and rectangular constraints. We also create a virtual freeform collimator, which can form any possible segment shape by opening or closing each bixel independently, to provide a basis for comparison. We formulate the problem of minimizing beam-on time as a large-scale linear programming problem. To deal with its dimensionality, we propose a column generation approach. We demonstrate the efficacy of our approach on a set of clinical problem instances. Our results indicate that the dual MLC under consecutiveness constraint yields very similar beam-on time to a virtual freeform collimator. Our approach also provides a ranking between other collimator technologies in terms of their delivery efficiencies.

Trust-region method for box-constrained semismooth equations and its applications to complementary problems

In this paper, we propose a new trust-region algorithm for bound-constrained semismooth systems of equations. Trust-region subproblem is defined by minimizing a quadratic function subject only to a rectangular constraint. By employing a new active set and nonmonotone techniques, solution of the equations can be found effective. Global and local convergence results of the proposed algorithm are established under reasonable conditions. The algorithm is applied and tested on complementary problems and the experiments show that our method is efficient.

Simultaneous Confidence Bounds in Multiple Regression Using Predictor Variable Constraints

Abstract This article presents a method for approximating the coverage probability of simultaneous confidence bounds for multiple regression functions when the vector of predictor variables is constrained to lie in a polyhedral convex set. The method is useful because it allows one to construct simultaneous confidence intervals with prescribed coverage probability for the regression function evaluated at various settings of the predictor variables, which are narrower than bounds obtained without using the predictor variable constraints. For a family of two-sided simultaneous confidence bounds that includes Scheffé-type and constant-width bounds, the probability of coverage is related to the distribution of the maximum Euclidean norm of the projection onto a polyhedral cone for a pair of random vectors with known joint distribution. An analogous relation holds for one-sided bounds. If an algorithm for computing the projection onto the cone is available, then these results enable one to use the Monte Carlo method to approximate the coverage probabilities of bounds. Particular attention is given to the case in which lower and upper bounds can be specified for each of the predictor variables so that the constraint region for bounding the regression function is rectangular. An efficient algorithm for calculating projections onto the appropriate cone for rectangular constraint regions, which facilitates coverage probability approximation of Schefffé-type bounds, is presented. This algorithm is used to calculate approximate critical points of Scheffé-type bounds for some specific designs and constraint regions and we show that substantial improvements for one-sided bounds and modest improvements for two-sided bounds over the conservative bounds of Casella and Strawderman (1980) may be obtained.