Generalized Method of Lagrange Multipliers: A Robust Optimization Strategy for Microwave Tomography

Optimization techniques in microwave tomography (MWT) suffer from convergence slowness and solution inaccuracy in problems associated with complex geometries and inhomogeneous media. By generalizing the method of Lagrange multipliers (MLM) in Newtonian framework, five steps are strategized to improve the convergence and accuracy of MWT. First, the gradient vectors of the objective and state equations are optimally parallelized. This is to guarantee that the achieved solution is optimum for both the objective and state equations in a biobjective optimization problem. Then, a systematic imposition of the lower and upper limits of the objective variables is allowed through an inequality constraint. This is to eliminate the echo and wave-like effects in the constructed solution. The other three steps are to retain computational time similar to the leading state-of-the-art methods. As such, the state and objective variables are dealt with independently; the solution of the state equation is linearly approximated, and the direct derivation of the computationally expensive second derivatives is avoided. The proposed strategy leads to at least one order of magnitude improvement in the convergence rate with respect to the state-of-the-art methods. Accordingly, the chance of confinement to false-positive or false-negative solutions is rigorously reduced. Verified examples include MWT for brain stroke imaging.