So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on:
Minimize the following expression (power dissipation):
$$I_{B1}(V - C_1) + I_{E2}(V - C_2) + I_{B2}(V - C_3) + I_{E3}(V - C_4) + I_{B3}(V - C_5) + I_{E4}(V - C_6) + I_{B4}(V - C7) + I_{X1}(V - C8) + I_{X2}(V - C_9) + I_{X3}(V - C_{10}) + I_{X4}(V - C_{11})$$
Subject to the following constraints:
$$I_{B1} + I_{L1} + I_{E2} = I_{L2} + I_{X1}$$ $$I_{B2} + I_{L2} + I_{E3} + I_{X1} = I_{L3} + I_{X2}$$ $$I_{B3} + I_{L3} + I_{E4} + I_{X2} = I_{L4} + I_{X3}$$ $$I_{B4} + I_{L4} + I_{X3} = I_{X4} + I_{L5}$$
Where: $$I_{L1} = A$$ $$I_{L2} = B$$ $$I_{L3} = C$$ $$I_{L4} = D$$ $$I_{L5} = E$$
Where $V, A, B, C, D, E$ and $C_1...C_{11}$ are all known constants and all variables and constants are real numbers.
Could anyone recommend a method for solving this problem (Lagrange multipliers)? Unfortunately I do not own a copy of Mathematica, is there another software package that can be used to solve systems like this?