Single Linear Constraint Research Articles

A Three-element Risley Prism Continuous Scanning System Based on Linear Constraint

In the three-element Risley prism system, non-linear errors, the infinite solution, the blind zone and the singularity are primary factors that affect the scanning accuracy of the system. In this paper, an accurate analytical model of the three-element Risley prism system is proposed. To reduce the errors of single-wedge nonlinear constraint equation, a dimensionality reduction method constrained by a single wedge is adopted. In addition, a reliable accuracy of 5mm is employed to achieve a full-area scanning of the system, and a single wedge linear constraint equation with an angular increment is established to control singularities. Simulation results show that the wedge rotation angle curve is stable and continuous in the process of inverse solution, and the full coverage of the scanning area can be achieved by assigning scanning accuracy to 5mm, which indicates that the proposed method can effectively solve the problems and achieve full-area scanning in the system.

A simplified completely positive reformulation for binary quadratic programs

In this paper, we propose a simplified completely positive programming reformulation for binary quadratic programs. The linear equality constraints associated with the binary constraints in the original problem can be aggregated into a single linear equality constraint without changing the feasible set of the classic completely positive reformulation proposed in the literature. We also show that the dual of the proposed simplified formulation is strictly feasible under a mild assumption.

Fast algorithm for quadratic knapsack problem

The paper considers a quadratic programming problem with a strictly convex separable objective function, a single linear constraint, and two-sided constraints on variables. This problem is commonly called the Convex Knapsack Separable Quadratic Problem, or CKSQP for short. We are interested in an algorithm for solving CKSQP with a linear time complexity. The papers devoted to this topic contain inaccuracies in the description of algorithms and ineffective implementations. In this work, the existing difficulties were overcome.

A new closed form multiple discrete-continuous extreme value (MDCEV) choice model with multiple linear constraints

Traditional multiple-discrete continuous choice models that have been formulated and applied in recent years consider a single linear resource constraint, which, when combined with consumer preferences, determines the optimal consumption point. However, in reality, consumers may face multiple resource constraints, such as those associated with time, money, and storage capacity. Ignoring such multiple constraints and instead using a single constraint can, and in general will, lead to poor data fit and inconsistent preference estimation, which can then have a serious negative downstream effect on forecasting and welfare/policy analysis. Unlike earlier attempts to address this multiple constraint situation, we formulate a new multiple-constraint (MC) multiple discrete continuous extreme value (MDCEV) model (or the MC-MDCEV model) that retains a closed-form probability structure and is as simple to estimate as the MDCEV model with one constraint. We achieve this by assuming a type-I extreme value distribution for the error term in its minimization form in the baseline utility preference of each good rather than a maximization form as in Bhat's (2005; 2008) original MDCEV formulation. The statistical foundation of the proposed model is based on the fact that the difference between a minimal type-I extreme value random variable with scale σ and the weighted sum of the exponential of standardized minimal type-I extreme value random variables (scaled up by σ) leads to an apparently new multivariate distribution that has an elegant and closed-form survival distribution function. Results from a simulation experiment show that our proposed model substantially outperforms single-constraint models; the results also highlight the serious mis-estimation that is likely to occur if only a subset of active constraints is used. The proposed model is applied to a case of week-long activity participation where individuals are assumed to maximize their utility from time-use subject to time and money budgets.It is hoped that our proposed simple closed-form multi-constraint MDCEV model will contribute to a new direction of application possibilities and to new research into situations where consumers face multiple constraints within a multiple discrete-continuous choice context.

An efficient gradient projection method for structural topology optimization

This paper presents an efficient gradient projection-based method for structural topological optimization problems characterized by a nonlinear objective function which is minimized over a feasible region defined by bilateral bounds and a single linear equality constraint. The specialty of the constraints type, as well as heuristic engineering experiences are exploited to improve the scaling scheme, projection, and searching step. In detail, gradient clipping and a modified projection of searching direction under certain condition are utilized to facilitate the efficiency of the proposed method. Besides, an analytical solution is proposed to approximate this projection with negligible computation and memory costs. Furthermore, the calculation of searching steps is largely simplified. Benchmark problems, including the MBB, the force inverter mechanism, and the 3D cantilever beam are used to validate the effectiveness of the method. The proposed method is implemented in MATLAB which is open-sourced for educational usage.

The convex hull heuristic for nonlinear integer programming problems with linear constraints and application to quadratic 0–1 problems

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

A Two-Phase Gradient Method for Quadratic Programming Problems with a Single Linear Constraint and Bounds on the Variables

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportioning, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

Coderivatives Related to Parametric Extended Trust Region Subproblem and Their Applications

This paper deals with the Fréchet and Mordukhovich coderivatives of the normal cone mapping related to the parametric extended trust region subproblems (eTRS), in which the trust region intersects a ball with a single linear inequality constraint. We use the obtained results to investigate the Lipschitzian stability of parametric eTRS. We also propose a necessary condition for the local (or global) solution of the eTRS by using the coderivative tool.

The performance of model averaged tail area confidence intervals

ABSTRACTWe investigate the exact coverage and expected length properties of the model averaged tail area (MATA) confidence interval proposed by Turek and Fletcher, CSDA, 2012, in the context of two nested, normal linear regression models. The simpler model is obtained by applying a single linear constraint on the regression parameter vector of the full model. For given length of response vector and nominal coverage of the MATA confidence interval, we consider all possible models of this type and all possible true parameter values, together with a wide class of design matrices and parameters of interest. Our results show that, while not ideal, MATA confidence intervals perform surprisingly well in our regression scenario, provided that we use the minimum weight within the class of weights that we consider on the simpler model.

A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality

In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the Euclidean ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method.

Trust region subproblem with an additional linear inequality constraint

This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with a single linear inequality constraint. We present an efficient algorithm to solve the problem using a diagonalization scheme that requires solving a simple convex minimization problem. Attainment of the global optimality conditions is discussed. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.

Simplified semidefinite and completely positive relaxations

This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented by Burer. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the completely positive or the semidefinite approximation. It also shows that a tightening of the semidefinite relaxation proposed by Burer is equivalent to the original relaxation.

Parametric approach for solving quadratic fractional optimization with a linear and a quadratic constraint

This paper studies a class of nonconvex fractional minimization problem in which the feasible region is the intersection of the unit ball with a single linear inequality constraint. First, using Dinkelbach’s idea, it is shown that finding the global optimal solution of the underlying problem is equivalent to find the unique root of a function. Then using a diagonalization technique, we present an efficient method to solve the indefinite quadratic minimization problem with the original problem’s constraints within a generalized Newton-based iterative algorithm. Our preliminary numerical experiments on several randomly generated test problems show that the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large-scale problems.

The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems

This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.

Fast optimization-based conservative remap of scalar fields through aggregate mass transfer

We develop a fast, efficient and accurate optimization-based algorithm for the high-order conservative and local-bound preserving remap (constrained interpolation) of a scalar conserved quantity between two close meshes with the same connectivity. The new formulation is as robust and accurate as the flux-variable flux-target optimization-based remap (FVFT-OBR) [1,2] yet has the computational efficiency of an explicit remapper. The coupled system of linear inequality constraints, resulting from the flux form of remap, is the main efficiency bottleneck in FVFT-OBR. While advection-based remappers use the flux form to directly enforce mass conservation, the optimization setting allows us to treat mass conservation as one of the constraints. To take advantage of this fact, we consider an alternative mass-variable mass-target (MVMT-OBR) formulation in which the optimization variables are the net mass updates per cell and a single linear constraint enforces the conservation of mass. In so doing we change the structure of the OBR problem from a global linear-inequality constrained QP to a singly linearly constrained QP with simple bounds. Using the structure of the MVMT-OBR problem, and the fact that in remap the old and new grids are close, we are able to develop a simple, efficient and easily parallelizable optimization algorithm for the primal MVMT-OBR QP. Numerical studies on a variety of affine and non-affine grids confirm that MVMT-OBR is as accurate and robust as FVFT-OBR, but has the same computational cost as the explicit, state-of-the-art FCR.

An affine scaling method for optimization problems with polyhedral constraints

Recently an affine scaling, interior point algorithm ASL was developed for box constrained optimization problems with a single linear constraint (Gonzalez-Lima et al., SIAM J. Optim. 21:361---390, 2011). This note extends the algorithm to handle more general polyhedral constraints. With a line search, the resulting algorithm ASP maintains the global and R-linear convergence properties of ASL. In addition, it is shown that the unit step version of the algorithm (without line search) is locally R-linearly convergent at a nondegenerate local minimizer where the second-order sufficient optimality conditions hold. For a quadratic objective function, a sublinear convergence property is obtained without assuming either nondegeneracy or the second-order sufficient optimality conditions.

Duality and Optimization for Generalized Multi-Hop MIMO Amplify-and-Forward Relay Networks With Linear Constraints

We consider a generalized multi-hop MIMO amplify-and-forward (AF) relay network with multiple sources/destinations and arbitrarily number of relays. We establish two dualities and the corresponding dual transformations between such a network and its dual, respectively under single network linear constraint and per-hop linear constraint. The result is a generalization of the previous dualities under different special cases and is proved using new techniques which reveal more insight on the duality structure that can be exploited to optimize MIMO precoders. A unified optimization framework is proposed to find a stationary point for an important class of non-convex optimization problems of AF relay networks based on a local Lagrange dual method, where the primal algorithm only finds a stationary point for the inner loop problem of maximizing the Lagrangian w.r.t. the primal variables. The input covariance matrices are shown to satisfy a polite water-filling structure at a stationary point of the inner loop problem. The duality and polite water-filling are exploited to design fast primal algorithms. Compared to the existing algorithms, the proposed optimization framework with duality-based primal algorithms can be used to solve more general problems with lower computation cost.

Resampling versus Repair in Evolution Strategies Applied to a Constrained Linear Problem

We study the behaviour of multi-recombination evolution strategies for the problem of maximising a linear function with a single linear constraint. Two variants of the algorithm are considered: a strategy that resamples infeasible candidate solutions and one that applies a simple repair mechanism. Integral expressions that describe the strategies' one-generation behaviour are derived and used in a simple zeroth order model for the steady state attained when operating with constant step size. Applied to the analysis of cumulative step size adaptation, the approach provides an intuitive explanation for the qualitative difference in the algorithm variants' behaviour. The findings have implications for the design of constraint handling techniques to be used in connection with cumulative step size adaptation.

Accommodating multiple constraints in the multiple discrete–continuous extreme value (MDCEV) choice model

Multiple-discrete continuous choice models formulated and applied in recent years consider a single linear resource constraint, which, when combined with consumer preferences, determines the optimal consumption point. However, in reality, consumers face multiple resource constraints such as those associated with time, money, and capacity. Ignoring such multiple constraints and instead using a single constraint can, and in general will, lead to poor data fit and inconsistent preference estimation, which can then have a serious negative downstream effect on forecasting and welfare/policy analysis.In this paper, we extend the multiple-discrete continuous extreme value (MDCEV) model to accommodate multiple constraints. The formulation uses a flexible and general utility function form, and is applicable to the case of complete demand systems as well as incomplete demand systems. The proposed MC-MDCEV model is applied to time-use decisions, where individuals are assumed to maximize their utility from time-use in one or more activities subject to monetary and time availability constraints. The sample for the empirical exercise is generated by combining time-use information from the 2008 American Time Use Survey and expenditure records from the 2008 US Consumer Expenditure Survey. The estimation results show that preferences can get severely mis-estimated, and the data fit can degrade substantially, when only a subset of active resource constraints is used.

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.