Feasible direction

Zoutendijk s method of feasible direction Cembrowicz and Harrington (C4)... 

Now consider Figure 8.4, in which the gradient vectors Vgx and Vg2 are drawn. Note that — V/is contained in the cone generated by Vgx and Vg2. What if this were not so If —V/ were slightly above Vg2, it would make an angle of less than 90° with a feasible direction just below the line x + y — 2 = 0. If — V/ were slightly below Vg1 it would make an angle of less than 90° with a feasible direction just... 

Even if it were possible to explain the cosmological redshift phenomenon by de Broglie s aging photon model, the practical feasible direct test of the model is not easy to perform because the minimum necessary distance for the aging effect to be noted is considerable. Let us make a rough estimate of this distance, assuming that the cosmological redshift is due only to the interaction of the photon with the subquantum medium. 

Definition 3.2.5 (Feasible direction vector) Let a feasible point S F. Then, any point jc in a ball of radius e around S which can be written as S + d is a nonzero vector if and only if jc S. A vector d 0 is called a feasible direction vector from S if there exists a ball of radius e ... 

The set of feasible direction vectors d 0 from x is called the cone of feasible directions of F at... 

Illustration 3.2.1 Figure 3.2 shows the feasible region, a point x, and feasible direction vectors from jc. 

Remark 2 If jc is a local minimum of (3.3) and d is a feasible direction vector from jc, then for sufficiently small A, we must have... 

Lemma 3.2.1 Let d be a non-zero feasible direction vector from jc. Then, x must satisfy the conditions ... 

Definition 3.2.6 (Improving feasible direction vector) A feasible direction vector d 0 at jc is called an improving feasible direction vector at x if... 

Figure 3.2 Feasible region and feasible direction vectors...

Remark 1 The corresponding geometric necessary optimality condition is that the set of feasible directions defined by... 

Even Figure 5 prompts some thoughts about the convenience of using thermodynamic variables. The form (topology) of the surface of function G (x) helps find the feasible directions of motion to the point G(xext), which maps the point F(xext) in the thermodynamic space. These directions are invariant with respect to the second law of thermodynamics and lead to the extremum of the characteristic thermodynamic function of the system (in case, shown in Figure 5, to the minimum G, i.e., to G(xeq)). 

Identifying constraints on reaction directions is essential for applications of metabolic flux analysis. However, in many applications the procedure used for determining reaction directions is not concretely defined. Typically, a subset of the reactions in a model is assigned as irreversible and the feasible directions are assigned based on information in pathway databases [59], In these applications, by treating certain reactions as implicitly unidirectional, biologically reasonable results can often be obtained without considering the system thermodynamics as outlined above. 

Construct the stoichiometric matrix for this system, given the reaction numbering defined in the figure. Assume that internal reactions 1 through 4 are irreversible with feasible directions indicated in the figure. Reactions 6 and 7 are transport reactions, also irreversible with directions indicated. If the maximum uptake rate of A is 1 (in arbitrary units), what is the maximal output of D Given that production of D is optimal, is the internal flux distribution unique ... 

Conceptually, methods of feasible directions operate in a manner similar to unconstrained multidimensional search techniques. That is, the basic idea is, given a feasible point Xj, determine a direction dj and a step length Oj that yield the new point Xj+, = Xj + a dj. In the constrained case, however, care must to taken to choose a direction dj that not only produces a point Xj+, that improves the objective function, but also maintains the feasibility of Xj+,. For a differentiable objective function /(x), an improving feasible direction dj at the point Xj 6 S, has the following two properties (see Figure 13) ... 

Thus, methods of feasible directions operate by determining an improving feasible direction and then solving a line search problem to determine the step length in that direction. This process is repeated until some stopping rule is satisfied. Since the sequence of points generated is feasible to the primal problem, these are often called primal methods. The way in which the directions are generated and the step sizes are computed determines the various methods. 

Figure 13 Illustration of a) Fea ble direction set at x, b) Improving direction set at x, and (c) Improving feasible direction set at x.

Zoutendijk, G. (1960), Methods of Feasible Directions, Elsevier, Amsterdam. 

Methods based on the search for d, to improve the objective function and, at the same time, fulfill the constraints are called Feasible Direction Methods. 

In past. Active Set Methods were overlapped to Feasible Direction Methods. Recently, other methods have been added to the Feasible Direction Methods category and do not adopt the technique used by the Active Set Methods to find d . For instance, they exploit a barrier on variable bounds and inequality constraints to consistently limit the search within the feasible region. 

There is a second reason why Feasible Direction Methods should not be referred to as methods that use the active constraints strategy the direction d is feasible only for linear constraints. 

There is a third reason why Feasible Direction Methods should not be called methods that use the active constraints strategy it is possible to ejqjloit the direction di also including the bound for the variables (beyond the bounds already existing) to limit the search region. In this case, the direction d is not used to perform a onedimensional search, since a Trust region method or Reduced-step method is used. 

The pros and cons of Active Set Methods that are also included in the Feasible Direction Methods or Reduced-step method category are dear. 

Toplds, D.M. and Veinott, A.F. (1967) On the convergence of some feasible direction algorithms for nonlinear programming. 

Several optimization algorithms, such as the feasible direction method, the gradient projection method, and the linearization method are coded in a multi-purpose program ADS (advanced Design Synthesis) [16], which was used in the present work. A modified feasible directions method [17], was selected in the present numerical example. 

G. Vanderplaats and F. Moses (1973) Structural Optimization by Methods of Feasible Directions , J. Comp. Struct., 3, 739-755. 

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