Optimization on this function will make the inner-divergence described in the first two terms on the right hand side as small as possible while the inter-class divergence among classes will be as big as possible, which will benefit the classification greatly. Different from the discriminative classifiers such as the LR, the discriminative information is finally incorporated into the joint probability p1 and p2. Thus the advantages of using joint probabilities will be naturally inherited into the discriminative Naive Bayesian classifier. However, the disadvantage of adding into this interactive item is that we cannot optimize p1 and p2 as in NB separately in the sub-dataset 1 and sub-dataset 2. To clarify this problem, we combine the NB assumption to expand the optimization function into a complete form: min {p1,p2} X 2 c=1 Xn j=1 X Aj [pˆc(ajk)log ˆ pc(ajk) pc(ajk) ] + W · 1 Pn j=1 P Aj p1(ajk)log(p1(ajk)/p2(ajk)),(10) s.t. 0 ≤ pc(ajk) ≤ 1, (11) X Aj pc(ajk) = 1, c = 1, 2;j = 1, 2, . . . n. (12) pc(ajk) is the short form of pc(Aj = ajk). So does pˆc(ajk). p1 and p2 are a set of parameters, namely, p1 = {p1(Aj ), 1 ≤ j ≤ n}, p2 = {p2(Aj ), 1 ≤ j ≤ n}. This is a nonlinear optimization problem under linear constraints. p1 and p2 are interactive variables. It is clear that they cannot be separately optimized as in Eq.( 7). To solve this problem, we use a modified Rosen’s Gradient Projection Method [16]. We firstly calculate the gradient of the optimization function w.r.t p1 and p2 as Eq. (13). We then project this gradient on the constraint plane. In our problem the projection matrix can be written as Eq. (16). The optimal step length α is searched in the projected gradient direction by using Quadratic Interpolation method [11]. The process is repeated until a local minimal is obtained. We write down the detailed steps as follows:
Gradient Projection Method for Nonlinear Programming
docs.lib.purdue.edu/cgi/viewcontent.cgi?article...
Purdue University
by LD Pyle - 1971 - Cited by 1 - Related articles
Jun 10, 1971 - Pyle, L. Duane, "A Simplex Algorithm - Gradient Projection Method for Nonlinear Programming" (1971). ... Frisch and Rosen are based on an interestingmethod for inverting ..... for example, by executing Phase I of the simplex.The Gradient Projection Method for Nonlinear Programming ...
www.jstor.org/stable/2098960
JSTOR
by JB Rosen - 1960 - Cited by 1256 - Related articles
the gradient projection method will also solve a linear programming prob- lem. In Part II of the paper ... 182 J. B. ROSEN in 1956 [15]. A number ... GRADIENT PROJECTION METHOD OF NONLINEAR PROGRAMMING 183 projection method ...[PDF]Gradient Projection Method for Nonlinear Programming
docs.lib.purdue.edu/cgi/viewcontent.cgi?article...
Purdue University
by LD Pyle - 1971 - Cited by 1 - Related articles
Jun 10, 1971 - ABSTRACT. Witzgall [ 7 L commenting on the gradient projection methods ... is generated using the simplex algorithm, whereas Rosen gives a.On Rosen's gradient projection methods - Springer
link.springer.com/.../10.1007%2FBF02...
Springer Science+Business Media
by DZ Du - 1990 - Cited by 24 - Related articles
This paper is a survey of Rosen's projection methods in nonlinear programming. Through the discussion of previous works, we propose some interesting ...The Gradient Projection Method for Nonlinear Programming ...
epubs.siam.org/.../0108011
Society for Industrial and Applied Mathematics
by JB Rosen - 1960 - Cited by 1256 - Related articles
The Gradient Projection Method for Nonlinear Programming. Part I. ... J. B. Rosen... (1982) Projected Newton Methods for Optimization Problems with Simple ...[PDF]Chapter 5: Constrained Optimization 5.5 Gradient Projection ...
www2.esm.vt.edu/~zgurdal/COURSES/4084/4084-Docs/.../GradProj.pdf
Rosen's gradient projection method is based on projecting the search .... Thegradient projection method has been generalized by Rosen to nonlinear con-.The Gradient Projection Method for Non-Linear ...
https://www.researchgate.net/.../250956400_The_Gradient_Pr...
The Gradient Projection Method for Non-Linear Programming, Part II. Non-Linear Constraints on ResearchGate, the professional network for ... J. B. Rosen.
ResearchGate
The Gradient Projection Method for Non-Linear ...
https://www.researchgate.net/.../240430418_The_Gradient_Pr...
The Gradient Projection Method for Non-Linear Programming, Part I. Linear ... which can be solved by using the gradient projection (GP) algorithm (Rosen, ...
ResearchGate
[PDF]Constrained Optimization 5 - Mechanical and Aerospace ...
www2.mae.ufl.edu/nkim/eas6939/ConstrainedOpt.pdf
Most problems in structural optimization must be formulated as constrained min ...... The gradient projection method has been generalized by Rosen to nonlinear[PPT]Engineering Optimization
www.zfm.ethz.ch/e/v/opt/handouts/ETHZ_Lecture10.ppt
Engineering Optimization – Concepts and Applications ... Rosen's gradient projection method; Zoutendijk's method of feasible directions ... nonlinearconstraints).
ETH Zurich
[PDF]CONSTRAINED NONLINEAR PROGRAMMING
www.pitt.edu/~jrclass/opt/notes4.pdf
We now turn to methods for general constrained nonlinear programming. These may ...... Rosen's method works by projecting -∇f(x) on to the .... The gradient projectionalgorithm can be generalized to nonlinear constraints by projecting the
University of Pittsburgh
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