Scientific contributions of Leo Khachiyan (a short overview)
Discrete Applied Mathematics
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
SIAM Journal on Optimization
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Let Y be a convex set in R/sup k/ defined by polynomial inequalities and equations of degree at most d/spl ges/2 with integer coefficients of binary length l. We show that if Y/spl cap/Z/sup k//spl ne//spl theta/, then Y contains an integral point of binary length ld/sup O/((k/sup 4/)). For fixed k, our bound implies a polynomial-time algorithm for computing an integral point y/spl isin/Y. In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming.