An exterior point simplex algorithm for (general) linear programming problems

We present an exterior point simplex type algorithm that possesses a new monotonic property. A dual feasible basic solution is required to start with. Intermediate solutions are neither primal nor dual feasible. Cycling-free pivoting rules and an exponentional example are presented.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

A simplex based algorithm to solve separated continuous linear programs

We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions. Research supported in part by US-Israel BSF grant 9400196, by German-Israel GIF grant I-564-246/06/97 and by Israel Science Foundation Grants 249/02 and 454/05.     

A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.This research has been supported in part by the National Science Foundation under grant DMS-8217261.     

A simplex algorithm for piecewise-linear programming II: Finiteness,feasibility and degeneracy

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for piecewise-linear programming, under convenient assumptions that guarantee a finite number of basic solutions, existence of basic feasible solutions, and nondegeneracy of all such solutions. Part II now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise-linear simplex algorithm. The theory of piecewise-linear programming is thereby extended, and numerous features of linear programming are generalized or are seen in a new light. An analysis of the algorithm's computational requirements and a survey of applications will be presented in Part III.This research has been supported in part by the National Science Foundation under grant DMS-8217261.     

A largest-distance pivot rule for the simplex algorithm

We propose a new pivot rule for the simplex algorithm, which is demonstrative in the dual space intuitively. Although it is based on normalized reduced costs, like the steepest-edge rule and its variants, the rule is much simpler and cheaper than the latter. We report computational results obtained with the 47 largest Netlib problems in terms of the number of rows and columns, all of the 16 Kennington problems, and the 17 largest BPMPD problems. Over the total 80 problems, a variant of the rule outperformed the Devex rule with iterations and time ratio 1.43 and 3.24, respectively.     

A variant of Karmarkar's linear programming algorithm for problems in standard form

This paper presents a variant of Karmarkar's linear programming algorithm that works directly with problems expressed in standard form and requires no a priori knowledge of the optimal objective function value. Rather, it uses a variation on Todd and Burrell's approach to compute ever better bounds on the optimal value, and it can be run as a prima-dual algorithm that produces sequences of primal and dual feasible solutions whose objective function values convege to this value. The only restrictive assumption is that the feasible region is bounded with a nonempty interior; compactness of the feasible region can be relaxed to compactness of the (nonempty) set of optimal solutions.     

On the mixed integer signomial programming problems

This paper proposes an approximate method to solve the mixed integer signomial programming problem, for which the objective function and the constraints may contain product terms with exponents and decision variables, which could be continuous or integral. A linear programming relaxation is derived for the problem based on piecewise linearization techniques, which first convert a signomial term into the sum of absolute terms; these absolute terms are then linearized by linearization strategies. In addition, a novel approach is included for solving integer and undefined problems in the logarithmic piecewise technique, which leads to more usefulness of the proposed method. The proposed method could reach a solution as close as possible to the global optimum.     

Convergence of a Dinkelbach-type algorithm in generalized fractional programming

The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.

---

Zusammenfassung Die Konvergenz eines Verfahrens i. S. von Dinkelbach zur Lösung verallgemeinerter Quotientenprogramme wird durch Untersuchung der Sensitivität eines parametrisierten Problems abgeleitet. Es wird gezeigt, daß die Konvergenzrate durch (1+5)/2 nach unten beschränkt ist, falls gewisse Regularitätsbedingungen in einer Umgebung der Optimallösung erfüllt sind. Ferner wird eine notwendige und hinreichende Bedingung zur quadratischen Konvergenz hergeleitet. Es wird gezeigt, wie diese im Falle konvexer Probleme implementiert werden kann.

     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

A new look at fractional programming

The parametric approach to fractional programming problems is examined and a new format is proposed for it. The latter reflects the fact that the approach as a whole capitalizes on a first-order necessary and sufficient optimality condition pertaining to differentiable pseudolinear functions.     

Simplex-inspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose objective function is given as a linear function plus a convex function whose arguments are linear functions of the decision variables and whose feasible region is a polytope. We show that there exists an optimal solution to this class of problems on a face of the constraint polytope of dimension not more than the number of arguments of the convex function. Based on this result, we develop a method to solve this problem that is inspired by the simplex method for linear programming. It is shown that this method terminates in a finite number of iterations in the special case that the convex function has only a single argument. We then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for an arbitrary number of arguments in the convex function. A computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables. The work of T. C. Sharkey was supported by a National Science Foundation Graduate Research Fellowship. The work of H. E. Romeijn was supported by the National Science Foundation under Grant No. DMI-0355533.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

A strongly polynomial simplex method for the linear fractional assignment problem

In this paper we show that the complexity of the simplex method for the linear fractional assignment problem (LFAP) is strongly polynomial. Although LFAP can be solved in polynomial time using various algorithms such as Newton’s method or binary search, no polynomial time bound for the simplex method for LFAP is known.     

A one-phase algorithm for semi-infinite linear programming

We present an algorithm for solving a large class of semi-infinite linear programming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on the constraints; it allows cuts that are not near the most violated cut; and it solves the primal and the dual problems simultaneously. We prove the convergence of this algorithm in two steps. First, we show that the algorithm can find an-optimal solution after finitely many iterations. Then, we use this result to show that it can find an optimal solution in the limit. We also estimate how good an-optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding an-optimal solution under some assumptions. This algorithm is generalized to solve a class of nonlinear semi-infinite programming problems. Applications to convex programming are discussed.     

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

A simple direct cosine simplex algorithm

Linear programming (LP) is the core model of constrained optimization. The Simplex method (Simplex in short) has been proven in practice to perform very well in small- or medium-sized LP problems. A new algorithm called the direct cosine Simplex algorithm (DCA) is presented here to improve upon Simplex and to solve LP problems. The proposed DCA implements a specific cosine criterion to choose the entering variable instead of the traditional most negative rule used in Simplex. Three examples are given to illustrate the implementation of the proposed DCA to improve Simplex and to serve as the optimization tool. The utility of the proposed approach is evident from the extensive computational results on test problems adapted from NETLIB. DCA reduced the number of iterations of Simplex in most cases in our computational experiment. Preliminary results for medium-sized problems are encouraging.