Murty, K. Carvajal-Moreno, R. Rockafellar, R. Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions.
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Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative. Skip to main content. Search SpringerLink Search. Abstract This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. References 1. Google Scholar 2.
Google Scholar 3. The information in the optimum tableau can be written explicitly as. A constraint equation can be used as a source row for generating a cut, provided its right-hand side is fractional.
We also note that the z-equation can be used as a source row because z happens to be integer in this example. We will demonstrate how a cut is generated from each of these source rows, starting with the z-equation. First, we factor out all the noninteger coefficients of the equation into an integer value and a fractional component, provided that the resulting fractional component is strictly positive. For example,. Moving all the integer components to the left-hand side and all the fractional components to the right-hand side, we get.
Because x 3 and x 4 are nonnegative and all fractions are originally strictly positive, the right-hand side must satisfy the following inequality:. The last inequality is the desired cut and it represents a necessary but not sufficient condition for obtaining an integer solution. It is also referred to as the fractional cut because all its co-efficients are fractions.
Thus, if we add this cut to the optimum tableau, the resulting optimum extreme point moves the solution toward satisfying the integer requirements.
Before showing how a cut is implemented in the optimal tableau, we will demonstrate how cuts can also be constructed from the constraint equations. Consider the x 1 -row:. Anyone of three cuts given above can be used in the first iteration of the cutting-plane a1gorithm.
It is not necessary to generate all three cuts before selecting one. Arbitrarily selecting the cut generated from the x 2 -row, we can write it in equation form as.
This constraint is added to the LP optimum tableau as follows:. The tableau is optimal but infeasible. We apply the dual simplex method Section 4. The last solution is still noninteger in x l and x 3. Let us arbitrarily select x l as the next source row-that is,. The dual simplex method yields the following tableau:. It is not accidental that all the coefficients of the last tableau are integers, a property of the implementation of the fractional cut. Abstract This paper addresses itself to a special class of nonconvex quadratic program referred to as a bilinear program in the literature.
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