Functionals Subject

The intuition suggests some things in an attempt to write down the locally one-dimensional (LO.S) scheme in conformity with available constructions of Section 3. In working on the segment 0 < t we introduce a grid - tj = jr, j = 0,1,... with step r = t /ja and involve arbitrary functions subject to the normalization condition... 

Given a set of data points (x y,), i=l,...,N and a mathematical model of the form, y = f(x,k), the objective is to determine the unknown parameter vector k by minimizing the least squares objective function subject to the equality constraint, namely... 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

Chapter 1 presents some examples of the constraints that occur in optimization problems. Constraints are classified as being inequality constraints or equality constraints, and as linear or nonlinear. Chapter 7 described the simplex method for solving problems with linear objective functions subject to linear constraints. This chapter treats more difficult problems involving minimization (or maximization) of a nonlinear objective function subject to linear or nonlinear constraints ... 

Luus, R. arid T. Jaakola. Optimization of Nonlinear Function Subject to Equality Constraints. Chem Process Des Develop 12 380-383 (1973). 

To apply the procedure, the nonlinear constraints Taylor series expansion and an optimization problem is resolved to find the solution, d, that minimizes a quadratic objective function subject to linear constraints. The QP subproblem is formulated as follows ... 

Beale, E. (1955) On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society, Series B (Methodological), 17, 173. 

Schizophrenic adolescents who are characterized by deficits in cognitive functioning, subjective complaints, and psychopathological symptoms do profit from a structured 1-year rehabilitation program. Most of these deficiencies improved considerably. 

In solving for the extremum of a general function / subject to the constraints g = constant and h = constant, we can use the Lagrange s method of undetermined multipliers. That is, we can solve for... 

Wilson and co-workers have also considered optimal control of molecular dynamics in the strong-field regime using the density matrix representation of the state of the system [32]. This formulation is also substantially the same as that of Kosloff et al. [6] and that of Pierce et al. [8, 9]. Kim and Girardeau [33] have treated the optimization of the target functional, subject to the constraint specified by (4.8), using the Balian-Veneroni [34] variational method. The overall structure of the formal results is similar to that we have already described. 

Nonlinear and Mixed-Integer Optimization addresses the problem of optimizing an objective function subject to equality and inequality constraints in the presence of continuous and integer variables. These optimization models have many applications in engineering and applied science problems and this is the primary motivation for the plethora of theoretical and algorithmic developments that we have been experiencing during the last two decades. 

Masuo et al. [19] measured renal function (creatinine, BUN, creatinine clearance) and plasma norepinephrine levels over a 5-year period in nonobese, normotensive men with normal renal function. Subjects who had a significant deterioration of renal function (>10% increases from baseline of creatinine and BUN or decrease in creatinine clearance) over a 5-year period had higher plasma norepinephrine at the entry period and greater increases in plasma norepinephrine over 5 years (Figure 2). In this study, subjects who had significant changes in body weight or blood pressure were excluded. 

The optimal control problem is to choose an admissible set of controls u(t), and final time, tF, to minimize the objective function, /, subject to the bounds on the controls and constraints. 

A process-synthesis problem can be formulated as a combination of tasks whose goal is the optimization of an economic objective function subject to constraints. Two types of mathematical techniques are the most used mixed-integer linear programming (MILP), and mixed-integer nonlinear programming (MINLP). 

F(c/c,) denotes the dimensionless form of an arbitrary rate expression./(x) is a nonuniform, normalized catalyst activity distribution inside the pellet. A(x) is an auxiliary function, subject to the following linear differential equation ... 

Nieber and Doltsinis [64] calculated 15 nonadiabatic surface hopping trajectories starting from configurations and velocities sampled from a ground state simulation. The nonradiative excited state lifetime has been determined by fitting the time-dependent, decaying excited state ensemble population to a mono-exponential function subject to the boundary condition that all molecules were in the Sj state... 

The solution of Equation A-3 for the Green s function, subject to the boundary conditions in Equations A-6c and A-6d, is ... 

Much of the interest in acetal chemistry is related to carbohydrates, because polyhydroxyaldehydes exist predominantly as cyclic hemiacetalsJ Digoxin, for example, possesses acetal functions subject to hydrolysis. 

This function is called the objective function. The decision variables may be independent, but they will usually be related to each other by many constraint equations. The optimization problem can then be stated as maximization or minimization of the objective function subject to the set of constraints. Constraint equations are discussed in the next section. 

For this case of study, it is supposed that the four states are directly measurable. The estimation problem is posed as a least squares objective function subject to the model nonlinear differential equations as constraints, restricting the mathematical program to the size of the moving window, and therefore ignoring the data outside such window. 

Variational determination of the C, coefficients in Eq. (7) without truncation is referred to as full Cl and is the exact wave function subject to approximations in the operator (e.g., nonrelativistic) and in the independent-particle reference (e.g., basis set limit). 

Lymphocyte function Subjects age and sex (n) EPA-DHA (g/day) Duration (wk) Reference study type... 

Mixed-integer linear programming (MILP) problems require maximizing or minimizing a linear function subject to linear equality or inequality constraints with integer restrictions on some or all the variables. The mathematical statement of MILP can be expressed as ... 

The principle of the decentralized set-up is that the network system is composed of several independent entities individually operated by self-interested parties. Each independent entity has its own profit function subject to its own processing or transportation constraints, and is not willing to reveal its own information to other entities or the public. Often the decision variables for each entity in a decentralized system are also influenced by other entities decisions. The foundations of the decentralized RPS models are derived from our recent work in a multitiered RPS network. Using this decentralized RPS network framework, we obtain the equilibrium collection fee paid by the upstream boundary tier site and the resulting material flow allocation, as well as the internal transaction prices within the network. 

Electroretinogram (ERG) The ERG is a device designed to measure the electrical response of the eye to brief, high intensity flashes. In this study, the ERG was used to determine the association between JP8 exposure and retinal Mueller cell function. Subjects who volunteered for this protocol underwent an ERG as part of their evaluation. Since the hypothesized retinal changes are associated with chronic JP8 exposure, the ERG procedure was accomplished only once on the enrollee selected. In addition to comparisons between the exposed and unexposed groups, ERG results were compared with normative data. 

Core algorithm, or functional subjected to optimization. This determines how solutions are found. 

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