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**11 - 13**of**13**### Criss-Cross Pivoting Rules

"... . Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' criss-cross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , then th ..."

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. Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' criss-cross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , then there are some primal or dual infeasible variables. One might choose any of these. It is advised to choose once a primal and then a dual infeasible variable, if possible. ffl If the selected variable is dual infeasible, then it enters the basis and the leaving variable is chosen among the primal feasible variables in such a way that primal feasibility of the currently primal feasible variables is preserved. If no such basis exchange is possible another infeasible variable is selected. ffl If the selected variable is primal infeasible, then it leaves the basis and the entering variable is chosen among th

### Recollections on the discovery of the reverse search technique

"... Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, suppo ..."

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Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, supported by a JSPS/NSERC bilateral exchange. Komei was then working at the University of Tsukuba, Otsuka, which was a couple of subway stations from my office. So we met quite often. One day Komei visited me at my office in Todai and explained to me the criss-cross method for solving linear programs, independently developed by S. Zionts [11], T. Terlaky [8, 9] and Z. Wang [10]. In this method one pivots in the hyperplane arrangement generated by the constraints of the linear program, without regard for feasibility- thus differentiating it from the simplex method. Komei and Tomomi Matsui, a Ph.D. student at Tokyo Institute of Technology, had developed an elegant new proof of the convergence of the criss-cross method [6], which Komei was explaining to me. Komei had drawn a line-arrangement on the blackboard, along with the path the criss-cross method would take from any given vertex to the optimum vertex of the LP. On the board all of these edges were shown in yellow with directions that eventually