Reduction Algorithm For Linear Programming Research Articles

Computational experience with a modified potential reduction algorithm for linear programming

We study the performance of a homogeneous and self-dual interior point solver for linear programming (LP) that is equipped with a continuously differentiable potential function. Our work is motivated by the apparent gap between the theoretical complexity results and long-step practical implementations in interior point algorithms. The potential function described here ensures a global linear polynomial-time convergence while providing the flexibility to integrate heuristics for generating the search directions and step length computations. Computational results on standard test problems show that LP problems are solved as efficiently (in terms of the number of iterations) as Mosek6 .

On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(√nL) iterations. Among them, several potential reduction algorithms combine both theoretical (O(√nL) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(√nL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least Θ(√nL) iterations to find an optimal solution.

Large step volumetric potential reduction algorithms for linear programming

We consider the construction of potential reduction algorithms using volumetric, and mixed volumetric — logarithmic, barriers. These are true “large step” methods, where dual updates produce constant-factor reductions in the primal-dual gap. Using a mixed volumetric — logarithmic barrier we obtain an $$O(\sqrt {nmL} )$$ iteration algorithm, improving on the best previously known complexity for a large step method. Our results complement those of Vaidya and Atkinson on small step volumetric, and mixed volumetric — logarithmic, barrier function algorithms. We also obtain simplified proofs of fundamental properties of the volumetric barrier, originally due to Vaidya.

A Complexity Analysis for Interior-Point Algorithms Based on Karmarkar’s Potential Function

A new complexity analysis for constant potential reduction algorithms for linear programming is considered. Using Karmarkar’s primal-based potential function, it is shown that for a class of linear programs the number of iterations required is at most $O( mt + n\log ( n ) )$, where m is the number of equations and n is the number of variables in the standard linear programming form, and t can be interpreted as the number of significant figures required in the final solution. It is also shown that this result holds in expectation for some randomly generated problems. At the same time, it is shown that Karmarkar’s algorithm requires at least $\Omega ( n )$ iterations to solve Anstreicher’s linear program, if the initial solution is poor but valid.

A partial updating algorithm for linear programs with many more variables than constraints

We present a modified version of Ye's potential reduction algorithm for linear programming. By using partial updating and incorporating scaling factors based on subsets of variables, we are able to solve linear programs having many more variables than constraints faster than comparable full updating algorithms. Our scheme for choosing scaling factors is motivated by certain asymptotic properties of interior point methods.

A Short-Cut Potential Reduction Algorithm for Linear Programming

As most interior point algorithms iterate, they repeatedly perform costly matrix operations, such as projections, on the entire constraint matrix. For large-scale linear programming problems, such operations consume the great majority of the computation time required. However, for problems where the number of variables far exceeds the number of constraints, operations over the entire constraint matrix are unnecessary. We will examine and extend decomposition techniques which greatly reduce the amount of work required by such interior point methods as the dual affine scaling and the dual potential reduction algorithms. In an effort to judge the practical viability of the decompositioning, we compare the performance of the dual potential reduction algorithm with and without decompositioning over a set of randomly generated transportation problems. Accompanying a theoretical justification of these techniques, we focus on the implementation details and computational results of one such technique.

Combining phase I and phase II in a potential reduction algorithm for linear programming

This paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality. The algorithm is closely related to a similar method of Anstreicher. The new features are that we use a two-dimensional programming problem to derive better lower bounds than Anstreicher, that our direction-finding subproblem treats phase I and phase II more symmetrically, and that we do not need an initial lower bound. Our method also allows for the generation of a feasible solution (so that phase I is terminated) during the course of the iterations, and we describe two ways to encourage this behavior.

On partial updating in a potential reduction linear programming algorithm of Kojima, Mizuno, and Yoshise

We consider partial updating in Kojima, Mizuno, and Yoshise's primal-dual potential reduction algorithm for linear programming. We use a simple safeguard condition to control the number of updates incurred on combined primal-dual steps. Our analysis allows for unequal steplengths in the primal and dual variables, which appears to be a computationally significant factor for primal-dual methods. The safeguard we use is a primal-dual Goldstein-Armijo condition, modified to deal with the unequal primal and dual steplengths.

Near boundary behavior of primal—dual potential reduction algorithms for linear programming

This paper is concerned with selection of theź-parameter in the primal--dual potential reduction algorithm for linear programming. Chosen from [n + $$\sqrt n $$ , ź), the level ofź determines the relative importance placed on the centering vs. the Newton directions. Intuitively, it would seem that as the iterate drifts away from the central path towards the boundary of the positive orthant,ź must be set close ton + $$\sqrt n $$ . This increases the relative importance of the centering direction and thus helps to ensure polynomial convergence. In this paper, we show that this is unnecessary. We find for any iterate thatź can be sometimes chosen in a wide range [n + $$\sqrt n $$ , ź) while still guaranteeing the currently best convergence rate of O( $$\sqrt n $$ L) iterations. This finding is encouraging since in practice large values ofź have resulted in fast convergence rates. Our finding partially complements the recent result of Zhang, Tapia and Dennis (1990) concerning the local convergence rate of the algorithm.

Extensions of the potential reduction algorithm for linear programming

In this work, we study several extensions of the potential reduction algorithm that was developed for linear programming. These extensions include choosing different potential functions, generating the analytic center of a polytope, and finding the equilibrium of a zero-sum bimatrix game.

Long steps in an O(n 3 L) algorithm for linear programming

We consider partial updating in Ye's affine potential reduction algorithm for linear programming. We show that using a Goldstein--Armijo rule to safeguard a linesearch of the potential function during primal steps is sufficient to control the number of updates. We also generalize the dual step construction to apply with partial updating. The result is the first O(n3L) algorithm for linear programming whose steps are not constrained by the need to remain approximately centered. The fact that the algorithm has a rigorous "primal-only" initialization actually reduces the complexity to less than O(m1.5n1.5L).

On the Continuous Trajectories for a Potential Reduction Algorithm for Linear Programming

For a possibly degenerate linear program minx{cTx; Ax = b, x ≥ 0} (A is an m × n real matrix, b ∈ Rm and c ∈ Rn), whose optimal value is 0, we study the limiting behavior of the trajectories of the family of vector fields Φq(x) ≡ −XPAX [Xc − q≡1(cTx) 1]. for all values of q ≥ n, where X is the diagonal matrix associated with x and PAX is the projection operator onto the null space of AX. A polynomial algorithm based on the directions Φq(x) has been presented by Gonzaga [6] when q = n or q = n + √n. We show that all trajectories of Φq(x) converge to the unique “center” of the optimal face of the given linear program. When this face consists of a unique vertex, it is shown that any trajectory of Φq(x) approaches this vertex along the same direction. When the optimal face consists of more than one point, we show that there is a threshold value τ > 0 such that: for q > τ, “most” of the trajectories of Φq(x) converge to the “center” tangentially to the optimal face and that the direction of approach of a trajectory of Φq(x) depends on the initial condition: for q < τ (q = τ), the trajectories of Φq(x) converge to the “center” along a unique direction (along several directions which depend on the initial condition) forming a positive angle with the optimal face.

A note on a potential reduction algorithm for LP with simultaneous primal-dual updating

Potential function reduction algorithms for linear programming and the linear complementarity problem use key projections p x and p s which are derived from the ‘double’ potential function, φ( x, s) = ø ln( x T s)− Σ j = 1 n ln( x j s j ), where x and s are primal and dual slacks vectors. For non-symmetric LP duality we show that the existence of s, y, x satisfying s = c − A T y, A x = b such that p x = (ϱ/x T s) X s − e and p s = (ϱ/x T s)S x − e yields simultaneous primal and dual projection-based updating during the process of reducing the potential function ø. The role of x, s in an O(√nL) simultaneous primal-dual update algorithm is discussed.

O( nρL)-iteration and O( n3L)-operation potential reduction algorithms for linear programming

In this paper we propose two potential reduction algorithms, which we call Algorithm 1 and Algorithm 2, for linear programming. Algorithm 1 has parameters θ in the potential function and β which determines a step size. Suppose that θ= n 1− σ and β=0.2 n 0.5- ϱ for 0.5⩽ θ⩽ ϱ⩽1. Then Algorithm 1 requires at most O( n ϱ L) iterations and O( n 3 L) arithmetic operations in total. If we take θ= n and β=0.2, Algorithm 1 is similar to Ye's primal form algorithm. Algorithm 2 has a parameter θ= n 1− σ for σ∈[0.5, 1]. It requires at most O( n σ L) iterations and O( n 3 L) arithmetic operations in total. If θ= n , Algorithm 2 is similar to the long step algorithm of Anstreicher and Bosch.

An O(n 3 L) potential reduction algorithm for linear programming

We describe a primal-dual potential function for linear programming: $$\phi (x,s) = \rho \ln (x^T s) - \sum\limits_{j = 1}^n {\ln (x_j s_j )} $$ whereρ⩾ n, x is the primal variable, ands is the dual-slack variable. As a result, we develop an interior point algorithm seeking reductions in the potential function with $$\rho = n + \sqrt n $$ . Neither tracing the central path nor using the projective transformation, the algorithm converges to the optimal solution set in $$O(\sqrt n L)$$ iterations and uses O(n 3 L) total arithmetic operations. We also suggest a practical approach to implementing the algorithm.

REMARKS ON POTENTIAL VERSUS BARRIER FUNCTION METHODS FOR LINEAR PROGRAMMING

We will point out some relations between potential and barrier function methods for linear programming. Then, based on the relations, we will show that the classical logarithmic barrier function method for linear programming can be adjusted so that it generates the optimal solution in O(√<n>L) iterations, where n is the number of variables and L is the data length. The method can be seen as a barrier function version of Ye's "An O(n^3L) potential reduction algorithm for linear programming".