Best linear unbiased prediction for linear combinations in general mixed linear models

The general mixed linear model can be written as . In this paper, we mainly deal with two problems. Firstly, the problem of predicting a general linear combination of fixed effects and realized values of random effects in a general mixed linear model is considered and an explicit representation of the best linear unbiased predictor (BLUP) is derived. In addition, we apply the resulting conclusion to several special models and offer an alternative to characterization of BLUP. Secondly, we recall the notion of linear sufficiency and consider it as regards the BLUP problem and characterize it in several different ways. Further, we study the concepts of linear sufficiency, linear minimal sufficiency and linear completeness, and give relations among them. Finally, four concluding remarks are given.     

Admissible linear estimators in restricted linear models

Necessary and sufficient conditions are established for a linear estimator to be admissible among the set of all homogeneous and inhomogeneous, linear estimators under a linear model with the vector of parameters subject to linear restrictions. These conditions are then utilized to characterize influence that restrictions involved in a linear model have on the class of admissible linear estimators.     

Uniform linear approximation in normed linear spaces

It has been shown that the theory of H-sets is important in the characterization of best uniform approximation of continuous real-or complex-valued functions. We here extend the theory of H-sets to the more general setting of functions with compact domain and with range contained in a Banach space. Using the definitions of H-sets, we construct a maximal linear functional and obtain inclusion theorems analogous to the classical case. It is then a simple matter to deduce a characterization of best approximation and show when uniqueness and strong uniqueness are achieved.     

On linear programs with linear complementarity constraints

The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and ℓ ₀ minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications. We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.     

Safe bounds in linear and mixed-integer linear programming

Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size. Mathematics Subject Classification (2000):primary 90C11, secondary 65G20     

The bilevel linear/linear fractional programming problem

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. In this paper we consider the bilevel linear/linear fractional programming problem in which the objective function of the first level is linear, the objective function of the second level is linear fractional and the feasible region is a polyhedron. For this problem we prove that an optimal solution can be found which is an extreme point of the polyhedron. Moreover, taking into account the relationship between feasible solutions to the problem and bases of the technological coefficient submatrix associated to variables of the second level, an enumerative algorithm is proposed that finds a global optimum to the problem.     

Algorithms for linear programming with linear complementarity constraints

Linear programming with linear complementarity constraints (LPLCC) is an area of active research in Optimization, due to its many applications, algorithms, and theoretical existence results. In this paper, a number of formulations for important nonconvex optimization problems are first reviewed. The most relevant algorithms for computing a complementary feasible solution, a stationary point, and a global minimum for the LPLCC are also surveyed, together with some comments about their efficiency and efficacy in practice.     

Are linear algorithms always good for linear problems?

We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.     

Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.     

Dense linear manifolds and application to linear differential operators

Oscillation criteria are obtained for a class of functional differential equations including the generalized Emden-Fowler equation $\text{x}{}^{\mathrm{(n)}}\text{+ q(t) ¦x[g(t)]¦}{}^{\mathrm{\gamma }}\text{sgn}\text{(x[g(t)]) = 0, γ > 0}$, as a special case. The purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear, sublinear, and linear differential equations with retarded arguments.     

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of randomfunctions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski??s method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as theminimizer of a stochastic penalized contrast function imitating Lepski??smethod among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers pointwise estimation as well as the estimation of local averages of the slope parameter.     

A canonical decomposition for linear operators and linear relations

An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear relation.     

A linear programming approach for linear multi-level programming problems

In an earlier paper, we proposed a modified fuzzy programming method to handle higher level multi-level decentralized programming problems (ML(D)PPs). Here we present a simple and practical method to solve the same. This method overcomes the subjectivity inherent in choosing the tolerance values and the membership functions. We consider a linear ML(D)PP and apply linear programming (LP) for the optimization of the system in a supervised search procedure, supervised by the higher level decision maker (DM). The higher level DM provides the preferred values of the decision variables under his control to enable the lower level DM to search for his optimum in a narrower feasible space. The basic idea is to reduce the feasible space of a decision variable at each level until a satisfactory point is sought at the last level.     

On approximation of approximately linear mappings by linear mappings

Assume A is a normed linear space, B is a Banach space, and f: A → B is a mapping “approximately linear.” We solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.”     

Solving linear program as linear system in polynomial time

A physically concise polynomial-time iterative-cum-non-iterative algorithm is presented to solve the linear program (LP) $Min{c}^{t}x\text{subject to}Ax=b,x\ge 0$. The iterative part–a variation of Karmarkar projective transformation algorithm–is essentially due to Barnes only to the extent of detection of basic variables of the LP taking advantage of monotonic convergence. It involves much less number of iterations than those in the Karmarkar projective transformation algorithm. The shrunk linear system containing only the basic variables of the solution vector $x$ resulting from $Ax=b$ is then solved in the mathematically non-iterative part. The solution is then tested for optimality and is usually more accurate because of reduced computation and has less computational and storage complexity due to smaller order of the system. The computational complexity of the combination of these two parts of the algorithm is polynomial-time $O\left(n^3\right)$. The boundedness of the solution, multiple solutions, and no-solution (inconsistency) cases are discussed. The effect of degeneracy of the primal linear program and/or its dual on the uniqueness of the optimal solution is mentioned. The algorithm including optimality test is implemented in Matlab which is found to be useful for solving many real-world problems.     

A sign-based linear method for horizontal linear complementarity problems

Numerical Algorithms - For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the...     

On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.