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The book presents a graduate level, rigorous, and self-contained introduction to linear optimization (LO), the presented topics beingexpressive abilities of LO; geometry of LO - structure of polyhedral sets, LO duality and its applications; traditional LO algorithms - primal and dual simplex methods, and network simplex method; polynomial time solvability of LO via ellipsoid algorithm; conic programming with emphasis on expressing abilities of second order and semidefinite optimization, and polynomial time primal-dual interior point algorithms for linear and semidefinite optimization.