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Let set S be defined by (x in En :f(x) <=c}

f: En -> E1 is convex and differentiable and gradient of f(xo) is not 0 when f(x) = c. Let xo be a point such that f(xo) = c and let d = gradient at xo. Let lamda be any positive number and define x1 = xo+ lamda(d)

we know:

||x1-xo|| <= ||x1-x||

suppse we have 2 islands:

f(x,y) <= c1 and g(x,y) <= c2

where f and g are contibuously differentiable and strictly convex functions whose gradient vectors are non zero when f(x,y) = c1 and g(x,y)=c2

We need to build a bridge between the two islands of shortest possible length. Using the result of the theorem of separating hyperplane, devise a search technique to find the optimal bridge location.

Test your routine on:

f(x,y)= x^2 +2y^2, c1=1

g(x,y)= (x-4)^2 + (y-3)^2, c2 = 1