Asymptotic analysis of the exponential penalty trajectory in linear programming

We consider the linear program rain { c'x: Ax ~< b } and the associated exponential penalty function fr(x) = c'x + rEexp [ ( A ~ bi)/r]. For r close to 0, the unconstrained minimizer x(r) offr admits an asymptotic expansion of the form x (r) = x* + rd* + *l(r) where x* is a particular optimal solution of the linear program and the error term ~(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory h(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis when r tends to ~: the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution. AMS Subject Classification: 90C05, 90C25, 90C31,49M30