Davide Cherubini - PhD Thesis - UniCA Eprints

6.1 Introductionfreedom) in the choice of x. In geometric terms, it can be shown that the optimalvalue of cx lies on the boundary of the polytope defined by the intersection of thehyperplanes Ax = b.Two families of solution techniques are widely used today, both visiting aprogressively improving series of trial solutions, until a solution is reached thatsatisfies the optimal conditions.The Simplex algorithm, devised by George B. Dantzig in 1947 [31], is analgebraic, iterative method that identifies an initial “basic solution” (called theCorner Point) and then systematically moves to an adjacent basic solution improvingthe value of the objective function. The algorithm starts and remains onthe boundary of the polytope, searching for an optimal point.Conceptually the method may be outlined in 5 steps:1. Determine a starting basic feasible solution setting (n − m) non-basic variablesto zero.2. Select an “entering” variable from the non-basic variables, which gives themaximum improvement of the value of the objective function. If none existsthen the optimal solution has been found.3. Select a “leaving” variable from the current basic variables and set it tozero (such variable becomes non-basic)4. Make the “entering” variable a basic variable and determine the new basicsolution5. Return to Step 1.The Interior-point method, by contrast, visits points within the interior ofthe feasible region. This method derives from techniques developed in the 1960sby Fiacco and McCormick for nonlinear programming [40], but its application tolinear programming dates back only to Karmarkar in 1984 [46].Karmarkar remarked the fact that moving through the interior of the feasibleregion of a linear programming problem, using the negative of the gradient of theobjective function as the movement direction, may trap the search into corners ofthe polytope. In order to avoid this “jamming”, the negative gradient is balanced43