Your problem can be formulated as a nonlinear programming problem in such way. Let $$f_i(x):= \text{piecewise} \left(\sum_{j=1}^{j=n} a_{i,j}x_j- b_i <0,0,1 \right). $$ Then we find $$ \max f(x)= \sum_{i=1}^{j=m}f_i(x) $$ under the constraint $$\sum_{j=1}^{j=n} x_j^2 =1. $$ This can be solved by global optimizers. I use the DirectSearch in Maple (See http://www.maplesoft.com/applications/view.aspx?SID=101333 .).

See an example in an *.mw file from http://rapidshare.com/files/1627770637/NP.mw .

Your problem can be formulated as a nonlinear programming problem in such way. Let $$f_i(x):= \text{piecewise} \left(\sum_{j=1}^{j=n} a_{i,j}x_j- b_i <0,0,1 \right). $$ Then we find $$ \max f(x)= \sum_{i=1}^{i=m}f_i(x) $$ under the constraint $$\sum_{j=1}^{j=n} x_j^2 =1. $$ This can be solved by global optimizers. I use the DirectSearch in Maple (See http://www.maplesoft.com/applications/view.aspx?SID=101333 .).

See an example in an *.mw file from http://rapidshare.com/files/1627770637/NP.mw .

Your problem can be formulated as a nonlinear programming problem in such way. Let $$f_j(x):= \text{piecewise} \left(\sum_{j=1}^{j=n} a_{i,j}x_j- b_i <0,0,1 \right). $$ Then we find $$ \max f(x)= \sum_{i=1}^{j=m}f_i(x) $$ under the constraint $$\sum_{j=1}^{j=n} x_j^2 =1. $$ This can be solved by global optimizers. I use the DirectSearch in Maple (See http://www.maplesoft.com/applications/view.aspx?SID=101333 .).

See an example in an *.mw file from http://rapidshare.com/files/1627770637/NP.mw .

Your problem can be formulated as a nonlinear programming problem in such way. Let $$f_i(x):= \text{piecewise} \left(\sum_{j=1}^{j=n} a_{i,j}x_j- b_i <0,0,1 \right). $$ Then we find $$ \max f(x)= \sum_{i=1}^{j=m}f_i(x) $$ under the constraint $$\sum_{j=1}^{j=n} x_j^2 =1. $$ This can be solved by global optimizers. I use the DirectSearch in Maple (See http://www.maplesoft.com/applications/view.aspx?SID=101333 .).

See an example in an *.mw file from http://rapidshare.com/files/1627770637/NP.mw .