General Linear Constraints Research Articles

Long-term thermal power planning at VEW ENERGIE using a multi-interval Bloom and Gallant method

Bloom and Gallant have proposed an elegant model for finding the optimal thermal schedule subject to matching the load duration curve and general linear constraints. Their method is based on a linear program with some linear equality constraints and many linear inequality constraints. There are well-documented specialized linear programming algorithms that are able to solve this problem. This paper proposes the extension of the Bloom and Gallant model to several successive intervals with constraints on generation encompassing different intervals. A procedure for finding an initial feasible point, which is an essential part of the solution procedure, is described here, together with details of the modeling of several operational constraints and computational results with real test cases.

A primal-dual trust-region algorithm for non-convex nonlinear programming

A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to general inequality and linear equality constraints. The method uses a primal-dual trust-region model to ensure descent on a suitable merit function. Convergence is proved to second-order critical points from arbitrary starting points. Numerical results are presented for general quadratic programs.

Blending curves for landing problems by numerical differential equations, I. Mathematical modelling

In [1], the blending curves can be modelled as the solutions of a system of boundary value problems of Ordinary Differential Equations (ODEs), to resemble a flexible elastic beam. Such ODE blending approaches are different from the traditional interpolation methods. The advantages of the ODE approaches are optimal blending curves obtained in minimum energy, and flexibility to complicated blending problems. Following the ideas in [1], this paper intends to model the blending curves of airplane landing problems, also by means of the ODE solutions. Since the airstrip is a bounded straight line EE′ , the landing point of an airplane (i.e., the blending curve) is just on line EE′ , instead of the fixed point discussed in [1]. It is due to the rather complicated boundary conditions that the existence and uniqueness of the ODE solutions should first be studied. In this paper, we have concluded that the ODE solutions are existential and unique if the flying direction BB′ at the beginning point of the blending curve is not parallel to the airstrip EE′ . Otherwise, the infinite solutions exist if the exterior force is also perpendicular to EE′ . Moreover, the mathematical modelling for other landing problems, antimissile problems, and the ODE system with general linear boundary constraints are discussed.

A Shifted-Barrier Primal-Dual Algorithm Model for Linearly ConstrainedOptimization Problems

In this paper we describe a Newton-type algorithm model for solving smooth constrained optimization problems with nonlinear objective function, general linear constraints and bounded variables. The algorithm model is based on the definition of a continuously differentiable exact merit function that follows an exact penalty approach for the box constraints and an exact augmented Lagrangian approach for the general linear constraints. Under very mild assumptions and without requiring the strict complementarity assumption, the algorithm model produces a sequence of pairs lx^k, λ^kr converging quadratically to a pair ({\bar{x},\bar\lambda}) where {\bar{x} satisfies the first order necessary conditions and {\bar\lambda} is a KKT multipliers vector associated to the linear constraints. As regards the behaviour of the sequence lx^kr alone, it is guaranteed that it converges at least superlinearly. At each iteration, the algorithm requires only the solution of a linear system that can be performed by means of conjugate gradient methods. Numerical experiments and comparison are reported.

A robust superlinearly convergent algorithm for linearly constrained optimization problems under degeneracy

In this paper, the problem of minimizing a convex function subject to general linear constraints is considered. An algorithm which is an extension of the method described in [4] is presented. And a new dual simplex procedure with lexicographic scheme is proposed to deal with the degenerative case in the sense that the gradients of active constraints at the iteration point are dependent. Unlike other methods, the new algorithm possesses the following important property that, at any iteration point generated by the algorithm, one can choose a set of the most suitable basis and from it one can drop all constraints which can be relaxed, not only one constraint once. This property will be helpful in decreasing the computation amount of the algorithm. The global convergence and superlinear convergence of this algorithm are proved, without any assumption of linear independence of the gradients of active constraints.

FIR filter designs with linear constraints using the eigenfilter approach

Conventionally, the eigenfilter approach has been successfully used to design a linear phase FIR filter without adding, any constraint, such as a low-pass filter, differentiator, and Hilbert transformer. When a constraint is incorporated in the eigenfilter design, only some special examples have been considered. The Nyquist filter is a typical example. In this paper, we first develop an eigenfilter approach to design a linear phase FIR filter in conjunction with general linear constraints. Then, we extend the proposed approach to the arbitrary FIR filter design with general linear constraints. Two design examples are presented to demonstrate the effectiveness of our approaches.

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints. In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.

Convergence Properties of an Augmented Lagrangian Algorithm for Optimization with a Combination of General Equality and Linear Constraints

We consider the global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems. In these methods, linear and more general constraints are handled in different ways. The general constraints are combined with the objective function in an augmented Lagrangian. The iteration consists of solving a sequence of subproblems; in each subproblem the augmented Lagrangian is approximately minimized in the region defined by the linear constraints. A subproblem is terminated as soon as a stopping condition is satisfied. The stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization. Our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints. In this paper, we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as show that potentially troublesome penalty parameters remain bounded away from zero.

Limited information estimation and testing subject to linear constraints

Recently LIML and 2SLS estimators have been derived under constraints on the reduced-form covariance matrix. However, the LIML estimator has not been derived under linear constraints on the structural coefficients. In this paper the LIML estimator is derived under general linear constraints and the efficiency of the derived estimator is compared with existing estimators. Likelihood ratio, Lagrange multiplier, Wald and Hausman test statistics are derived for testing the adequacy of the linear constraints. 2SLS versions of estimation and testing follow naturally and the same 2SLS test statistics are derived using the auxiliary regression approach. A similar technique is applied to full information estimation and testing.

Linear multiplicative programming

An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.

An Interior-Point Algorithm for Linearly Constrained Optimization

This paper describes an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the “projection” at each iteration is performed by using a primal–dual interior-point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a well-defined way. Higher-order derivative information on the manifold defined by the apparently active constraints can be used to increase the rate of local convergence.

A quadratic-tensor model algorithm for nonlinear least-squares problems with linear constraints

A new algorithm is presented for solving nonlinear least-squares and nonlinear equation problems. The algorithm is based on approximating the nonlinear functions using the quadratic-tensor model proposed by Schnabel and Frank. The problem statement may include simple bounds or more general linear constraints on the unknowns. The algorithm uses a trust-region defined by a box containing the current values of the unknowns. The objective function (Euclidean length of the functions) is allowed to increase at intermediate steps. These increases are allowed as long as our predictor indicates that a new set of best values exists in the trust-region. There is logic provided to retreat to the current best values, should that be required. The computations for the model-problem require a constrained nonlinear least-squares solver. This is done using a simpler version of the algorithm. In its present form the algorithm is effective for problems with linear constraints and dense Jacobian matrices. Results on standard test problems are presented in the Appendix. The new algorithm appears to be efficient in terms of function and Jacobian evaluations.

On Energy Rate Theorems for Linear First-Order Nonholonomic Systems

Starting with the Boltzmann/Hamel equations of motion of mechanical systems subject to general linear and first-order nonholonomic and rheonomic constraints, and in nonholonomic coordinates, this paper derives the general (reactionless) energy rate, or power, equation for such systems, in a straightforward and physically clear fashion. A power (reaction-containing) equation for these systems, but in holonomic coordinates, is also derived for completeness. An example of a sphere rolling on a uniformly spinning plane serves to illustrate these power theorems. The paper extends and corrects the recent work by Kane and Levinson (1988) on the same topic.

The efficient set mathematics when mean-variance problems are subject to general linear constraints

In this paper we develop the efficient set mathematics for the case where mean-variance portfolio problems are subject to general linear constraints. We derive closed-form expressions for the optimal portfolio, the associated multipliers, and the portfolio's expected return and variance in each interval where the active set changes; identify where kinks on the efficient frontier occur; and show that securities plot on a security market hyperplane. The analysis extends our knowledge of portfolio theory, clarifies the zero-beta Capital Asset Pricing Model and the conditions under which securities plot on the Security Market Line (SML), and provides insight into the ambiguities associated with using the SML criterion to measure investment performance.

Asymptotic results for a conditional Poisson log-linear model

A minimum modified transformed chi-square estimator for the conditional Poisson log-linear model subject to general linear exact and stochastic constraints is defined. Lagrange multiplier and Wald tests are given for the general exact linear hypothesis and the general stochastic linear hypothesis. The asymptotic distributions of the estimator and test statistics are presented.

An algorithm for maximum entropy image reconstruction from noisy data

A Maximum Aposteriori Probability (MAP-) approach is considered for reconstructing an image from noisy projections. This approach needs, according to Bayes rule, two specified distributions. The entropy concept is used to specify the à priori distribution of the unknown image vector. The second distribution (which reflects measurement errors) is taken as an uncorrelated normal distribution. The MAP approach then leads to a discrete, strictly convex, nonquadratic optimization problem. The model contains a real parameter which controls the degree of smoothing imposed on the solution. We propose the use of cross-validation for picking the value of this parameter. We derive an algorithm to solve the optimization problem based on Bregman's convex programming scheme. First a simple generalization of Bregman's method is made which leads to increased efficiency for general linear constraints. Based on this generalization we write down the corresponding Bregman method for our reconstruction problem. The method requires that in each iteration a single nonlinear equation is solved. We propose to use one step of Newton's method to approximate the root. Finally a convergence proof is given for the new algorithm.

Duality in constrained location problems

The dual of a facility location problem with general norms, distance constraints, and linear constraints is formulated.

On weak continuity and the Hodge decomposition

We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwell’s equations. This result was derived independently by Tartar who stated it in a recent survey article [7]. Our proof is explicit and uses the Hodge decomposition.

Broad-band beamforming and the generalized sidelobe canceller

This correspondence identifies general conditions under which the generalized sidelobe canceller (GSC) structure implements a linearly constrained minimum variance (LCMV) broad-band beam-former. In previously published work, conditions have been presented for the case in which the linear constraint is restricted to fixed gain in the steer direction. Here, that work is extended to include general linear constraints. A geometric interpretation is presented to provide insight into the relationship between the GSC structure and LCMV beamformers, and several recent developments in broad-band beam-forming are reviewed and related to the GSC structure.

Linear Least Squares with Bounds and Linear Constraints

An algorithm is given for solving linear least squares systems of algebraic equations subject to simple bounds on the unknowns and (more general) linear equality and inequality constraints. The method used is a penalty function approach wherein the linear constraints are (effectively) heavily weighted. The resulting system is then solved as an ordinary bounded least squares system except for some important numerical and algorithmic details. This report is a revision of an earlier work. It reflects some hard-won experience gained while using the resulting software to solve nonlinear constrained least squares problems.