Initial Feasible Point Research Articles

Optimisation de procédés chimiques complexes: Etude d'un atelier de monochloration du benzène

Several procedures for the computer aided design of complex plants involving chemical reactors, rectification columns and recycle streams are presented. In the first part, these cyclic processes are represented by a sequential, two-hierarchical-level tree. A “branch and bound” method is then used to find the optimal structure for given operating conditions. The synthesis of a benzene chlorination process with one or several reactors is used to illustrate the implementation of the algorithm. The structure and the operating conditions are both optimized in the second part of this work. By using structural parameters to represent physical unit connections within a superstructure, the problem becomes one of large-scale nonlinear programming (NLP). A projected reduced gradient algorithm, extended to nonlinear constrained problems, is used to solve the nonlinear program. The same example as was used in the first part shows the complementary nature of the two approaches, with the first one providing a good initial feasible point for the second.

An experimental investigation of a new multiplex method for linear programming

Karmarkar’s recent internal and iterative method for linear programming problems has resulted in a renewed interest in some older alternatives, other than the simplex method. Here a new multiplex and geometric method, which has some features in common with the older methods, is proposed and implemented. In this method the solution is found by following a gradient path through the interior of the feasible region and through subspaces of reduced dimension corresponding to the bounding hyper-surfaces of the feasible region. The path moves from an initial feasible point through a sequence of linear steps to a vertex of the polytope defined by the constraints. Although similar, the current method differs fundamentally from Rosen’s gradient projection method in that the successive search directions are obtained from the gradients of reduced problems of lesser dimension. These directions, when translated to the original space, do not necessarily correspond to the gradient projection directions. Once a vertex has been reached the new algorithm determines whether or not it is optimal by applying a simple perturbation procedure for which the perturbed points are generated as a by-product of the computed path to the vertex. If not optimal the algorithm proceeds by restarting from a perturbed point (on a suitable edge) with increased function value and the path is continued until the next vertex is reached. It is shown that the results of this experimental investigation of this method is promising since it has successfully been applied to a variety of test problems.

Discrete and continuous approaches to the optimal synthesis of distillation sequences

Recent developments in integer programming and in large-scale nonlinear programming allow the optimization of certain classes of chemical engineering plants. In this study two approaches for synthesizing distillation processes are presented and compared. In the discrete approach, an integer programming problem resulting from simplifying assumptions is solved with a branch-and-bound algorithm. In the continuous approach a large-scale nonlinear programming problem, whose formulation is rather difficult, is solved by means of a reduced-gradient algorithm. An example shows the complementary nature of the two approaches, the discrete approach providing a good initial feasible point for the continuous approach.

Decomposition strategy for designing flexible chemical plants

AbstractOne of the main computational problems faced in the optimal design of flexible chemical plants with multi‐period operation is the large number of decision variables that are involved in the corresponding nonlinear programming formulation. To overcome this difficulty, a decomposition technique based on a projection‐restriction strategy is suggested to exploit the block‐diagonal structure in the constraints. Successful application of this strategy requires an efficient method to find an initial feasible point, and the extension of current equation ordering algorithms for adding systematically inequality constraints that become active. General trends in the performance of the proposed decomposition technique are presented through an example.

Solving Systems of Linear Equations and Inequalities

In this paper we present an algorithm which determines whether or not a system of linear equations and inequalities is consistent and if consistent, calculates a solution (feasible point) to it. This problem often occurs as a subproblem to a constrained optimization problem when the technique used to solve the optimization problem requires an initial feasible point. Also included is the convergence analysis, numerical examples and the results of tests comparing our algorithm with other methods for solving this problem.

Combined phase I—phase II methods of feasible directions

This paper presents several new algorithms, generalizing feasible directions algorithms, for the nonlinear programming problem, min{f 0 (z) ∣f j (z) ≤ 0,j = 1, 2, ⋯ ,m}. These new algorithms do not require an initial feasible point. They automatically combine the operations of initialization (phase I) and optimization (phase II).