An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems

With many applications in engineering and in scientific fields, quantifier elimination (QE) has been attracting more attention these days. Cylindrical algebraic decomposition (CAD) is used as a basis for a general QE algorithm. We propose an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm for solving polynomial optimization problems. The main ideas are a bounded CAD construction approach and utilization of sign information. The bounded CAD constructs CAD only in restricted admissible regions to remove redundant projection factors and avoid lifting cells where truth values are constant over the region. By utilization of sign information we can avoid symbolic computation in the lifting phase. Techniques for implementation are also presented. These techniques help reduce the computing time. We have examined our implementation by solving many example problems. Experimental results show that our implementation significantly improves efficiency compared to our previous work.

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