Nonbasic Variables Research Articles

Redundancy elimination with a lexicographic solved form

Detection of redundant linear constraints is important for a variety of applications including linear programming, variable elimination, canonical forms, and convex hull algorithms to name a few. Systematic semantic methods (i.e. optimization) exist to identify all redundant constraints but they are generally computationally expensive. It is thus desirable to identify syntactic criteria to reduce the need for semantic methods. This paper presents new syntactic criteria to detect redundant and non-redundant constraints. The results exploit a lexicographic solved form for linear programming (originally proposed to detect implicit equalities) which makes it possible to define syntactic criteria much stronger than those proposed in the past. In particular, one of our results shows that a system in a lexicographic solved form with n constraints and m non-basic variables has at most min(2n-1,n+m) redundant variables. In addition, the paper presents new syntactic criteria to reduce the size of the constraint system before further applications of syntactic and semantic methods. These theoretical results have been integrated in a complete system to detect redundant linear constraints. Experimental evaluation shows that our syntactic criteria decide the redundancy of about 90 constraints and may produce orders of magnitude improvement in efficiency over previous work whose criteria were deciding between 24 and 56 increase with the size and with the sparsity of the constraint system and are especially dramatic when the number of variables is large compared to the number of constraints.

Partial Affine-Scaling for Linearly Constrained Minimization

We propose an interior point method for finding a stationary point of a nonlinear program with linear equality and nonnegativity constraints. This method maintains a basis at each iteration and updates the iterate by taking an affine-scaling step in the space of nonbasic variables. In the general case, we propose to choose the basis according to a rule that maximizes the basic components of the iterate. In the case where the feasible region is the product of simplices, we propose an alternative rule that minimizes the basic components of the cost gradient. We analyze the convergence of the method under each rule. A key feature of this method is that it can be implemented much like the simplex method.

Block pivot methods for solving frictional contact problems

Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is “relative small”. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables.

The column subtraction algorithm: An exact method for solving weighted set covering, packing and partitioning problems

We present a new and conceptually simple approach for finding exact solutions to large set covering, packing and partitioning problems. The proposed algorithm is based on a new method, called the column subtraction (or row sum) method. First, a linear programming relaxation of the set problem is solved to optimality, using a specialized version of the pivot and probe algorithm of Sethi and Thompson, which has proved to be very effective in avoiding the massive degeneracy typical of these problems. Then, the column subtraction branch and bound method, which proceeds by subtracting certain non-basic columns of the final simplex tableau, one at a time from the right hand side column, is used to produce optimal integer solutions. The algorithm begins with a novel greedy heuristic that makes use of only the nonbasic surplus variables for the column subtractions, and which generates excellent bounds quickly. Computational comparisons, based upon problems having up to 9000 variables and 800 constraints, highlight the effective overall performance of the column subtraction algorithm.

On solution-containing ellipsoids in linear programming

Ellipsoids that contain all optimal dual slack solutions and those that contain all optimal primal solutions and that are independent of the algorithm used are derived. Based upon these ellipsoids, two criteria each for detecting optimal basic and nonbasic variables prior to optimality in interior-point methods are obtained. Using these results, we then derive a sufficient condition for a linear program to be feasible.

Computational behavior of a feasible direction method for linear programming

We discuss a finite method of feasible directions for linear programs. The method begins with a BFS (basic feasible solution) and constructs a profitable direction by combining the updated columns of several nonbasic variables eligible to enter. Moving in this direction as far as possible, while retaining feasibility, leads to a point which is not in general a basic solution of the original problem, but corresponds to a BFS of an augmented problem with a new column. So this is called an interior move or a column adding move. Next we can either carry another interior move, or a reduction process which starts with the present feasible solution and leads to a BFS of the original problem with the same or better objective value. We show that interior moves and reduction processes can be mixed in many ways leading to different methods, all of which can be implemented by maintaining the basis inverse or a factorization of it. Results of a computational experiment are presented.

Resolving Degeneracy in the Generalized Reduced Gradient Method

The Generalized Reduced Gradient (GRG) method has proven to be one of the more robust and efficient algorithms currently available for solving nonlinear programming problems. The method divides the vector of design variables into two classes, nonbasic and basic variables, and employs the implicit function theorem to formulate a reduced, unconstrained problem in the nonbasic variables. In order to employ the implicit function theorem two assumptions are made: (1) The Jacobian matrix of the active constraints with respect to the basic variables is nonsingular; and (2) All basic variables are within their respective bounds. When the second condition is not satisfied then the current point is degenerate and further progress is not assured. Methods based on performing basis changes exist for resolving degeneracy. In this paper, we will describe a technique based on the method of feasible directions which has the advantage of requiring no basis changes to generate a new direction.

A variation on Karmarkar’s algorithm for solving linear programming problems

The algorithm described here is a variation on Karmarkar's algorithm for linear programming. It has several advantages over Karmarkar's original algorithm. In the first place, it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function. The minimum value of the objective function does not have to be known in advance. Secondly, in the absence of degeneracy, the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero. This makes it possible to identify an optimal basis before the algorithm converges.

On diagonally preconditioning the truncated Newton method for super-scale linearly constrained nonlinear programming

We present an algorithm for super-scale linearly constrained nonlinear programming (LCNP) based on Newton's method. In large-scale programming solving the Newton equation at each iteration can be expensive and may not be justified when far from a local solution. For super-scale problems, the truncated Newton method (where an inaccurate solution is computed by using the conjugate-gradient method) is recommended; a diagonal BFGS preconditioning of the gradient is used, so that the number of iterations to solve the equation is reduced. The procedure for updating that preconditioning is described for LCNP when the set of active constraints or the partition of basic, superbasic and nonbasic (structural) variables have been changed.

Some branch‐and‐bound procedures for fixed‐cost transportation problems

AbstractIn this article we present three properties that will improve the performance of branch‐and‐bound algorithms for fixed‐cost transportation problems. By applying Lagrangian relaxation we show that one can develop stronger up and down penalties than those traditionally used and also develop a strengthened penalty for nonbasic variables. We also show that it is possible to “look ahead” of a particular node and determine the solution at the next node without actually calculating it. We present computational evidence by comparing our developments with existing procedures.

Solving staircase linear programs by the simplex method, 2: Pricing

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The preceding paper considered ‘inversion’ routines that factorize the basis and solve linear systems. The present paper examines ‘pricing’ routines that compute reduced costs for nonbasic variables and that select a variable to enter the basis at each iteration. Both papers describe extensive (although preliminary) computer experiments, and can point to some quite promising results. For pricing in particular, staircase computation strategies appear to offer modest but consistent savings; staircase selection strategies, properly chosen, may offer substantial savings in number of iterations, time per iteration, or both.

Compact formulation of pipe network problems

Solution of problems in water distribution networks usually proceeds in one of two ways: (1) analysis of flows in the links, or (2) analysis of the head at the nodes. In the former case, which is somewhat commoner and has certain computational advantages, the user is usually required to generate input data by defining the set of independent loops in the circuited network in addition to the usual node and link characteristics. Such methods are acceptable for analysis where the topology of the network remains constant. A method is presented here that allows the node and loop equations to be generated automatically from a very simple data file defining nodal stipulations and link characteristics. The equations are reduced to a small set of simultaneous equations involving only the nonbasic or redundant variables. This greatly reduces the memory requirements and also speeds solution of the equations. Either an iterative linear approximation or the Newton–Raphson method may be used to solve the simultaneous equations. The approach is useful in design and optimization studies since it allows links to be deleted and/or restored with automatic generation of the necessary equations. The method may also be used with microcomputers or even pocket computers since the order of the system of equations to be solved is reduced to the number of loops in the system.

Methods of parametric non-linear programming

Abstract This paper deals with general parametric non-linear programming problems. First, two algorithms are presented to obtain a parametric optimal solution of the problem having a single parameter by reducing it successively to associated problems which contain a smaller number of variables. The reduction is accomplished by partitioning the variables into basic and non-basic, and also by generating a smaller problem from the non-basic variables only. It is shown that the two algorithms are essentially equivalent to each other. Next, this idea is extended to handle a certain class of multi-parametric problems. Finally, computational results of the algorithm are given.

Block pivot methods for solving the complementarity problem

A family of algorithms which solve the complementarity problem (and certain generalized complementarity problems) is introduced. In these algorithms block pivots (multiple exchanges of basic and nonbasic variables) are permitted. The geometry of complementarity points is studied. The proof that these algorithms converge in a finite number of steps is based on certain elementary results from group theory, and does not rely on monotonicity arguments.

Technical Note—A Note on Gomory Fractional Cuts

This note shows that the Gomory cut developed for the mixed integer program, when extracted from a source row in which the nonbasic variables are integer constrained, is stronger than the one derived for the integer program.

Technical Note—An Improved Branch-and-Bound Method for Integer Programming

This note proposes two extensions of the successful Beale and Small branch-and-bound mixed-integer algorithm. The integer requirements on nonbasic variables are utilized to calculate stronger “penalties” when searching down the solution tree and to give a stronger criterion for abandoning unprofitable branches of the tree when backtracking. This stronger criterion is obtained by making use of Gomory cutting-plane constraints. These modifications have produced considerable reductions of the searching effort required for pure integer and predominantly integer problems, and have the further advantage of being very easy to incorporate.

Classical Derivation of the Necessary and Sufficient Conditions for Optimal Linear Programs

This paper presents a new derivation of the key ideas of Dantzig's simplex algorithm by using the differential approach to nonlinear programming. Specifically, the necessary condition shows that a candidate for the optimum is a basic feasible solution, while the sufficiency condition indicates that the optimality indicator (cj − zj) for nonbasic variables xj must be nonpositive (nonnegative) in order for a maximization (minimization) linear program to be optimal.

A finiteness proof for modified dantzig cuts in integer programming

Let be a basic solution to the linear programming problem subject to: where R is the index set associated with the nonbasic variables. If all of the variables are constrained to be nonnegative integers and xu is not an integer in the basic solution, the linear constraint is implied. We prove that including these “cuts” in a specified way yields a finite dual simplex algorithm for the pure integer programming problem. The relation of these modified Dantzig cuts to Gomory cuts is discussed.

A note on cycling in the simplex method

AbstractAlthough cycling in the simplex method has long been known, a number of theoretical questions concerning cycling have not been fully answered. One of these, stated in [3], is to find the smallest example of cycling, and Beale's example with three equations and seven variables is conjectured to be the smallest one. The exact bounds on dimensions of cycling examples are established in this paper. We show that Beale's example is the smallest one which cycles at a non‐optimal solution, that a smaller one can cycle at the optimum, and that, in general (including the completely degenerate case), a cycling example must have at least two equations, at least six variables, and at least three non‐basic variables. Examples and geometries are given for the extreme cases, showing that the bounds are sharp.

Jacobians in Constrained Nonlinear Optimization

In constrained optimization problems the rates of change of the objective function with respect to the independent (nonbasic) variables have been interpreted as shadow prices, dual variables, and Lagrange multipliers. These derivatives are shown here to be ratios of Jacobians of the transformations between certain sets of variables. This result extends the Kuhn-Tucker conditions of nonlinear programming to optimum seeking problems in which information about the objective and constraint functions can be obtained only by direct measurement, the functions not being given in closed analytic form. Other derivatives helpful for maintaining feasibility during a search are also shown to be ratios of Jacobians. An example shows how these results might be incorporated into algorithms for seeking a constrained optimum.