Finding all solutions of nonlinearly constrained systems of equations

Costas D. Maranas, Christodoulos A. Floudas

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171 Scopus citations


A new approach is proposed for finding all ε-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All ε-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite ε-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in αBB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.

Original languageEnglish (US)
Pages (from-to)143-182
Number of pages40
JournalJournal of Global Optimization
Issue number2
StatePublished - Sep 1 1995

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics


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