Describing convex semialgebraic sets by linear matrix inequalities Markus Schweighofer Université de Rennes 1 International Symposium on Symbolic and Algebraic Computation Korea Institute for Advanced Study, Seoul July 28-31, 2009 Introduction n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by linear matrix inequalities n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by Linear Matrix Inequalities n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LinearMatrix Inequalities n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LinearMatrix Inequalities n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Describing convex semialgebraic sets by LMIs n n A semialgebraic set in R is a subset of R defined by a boolean combination of polynomial inequalities. In other words, semialgebraic sets are the sets defined by quantifier-free formulas inductively built up from polynomial inequalities by f:; ^; _g. If one allows for formulas combining polynomial inequalities by f:; ^; _; 8x 2 R; 9x 2 Rg, then the defined sets are still semialgebraic. In fact, given a formula ', one can compute a quantifier-free formula defining the same set by Tarski’s real quantifier elimination. If ' has only rational coefficients, then the same can be assured for . Modern algorithms use cylindrical algebraic decomposition. Chris Brown et al., Wednesday, Room B, 14:00 – 15:15 , without of loss of generality an existential one (since 8x 2 R : ' is equivalent to :9x 2 R : :'). n+m If ' defines S ⊆ R , then 9xm+1;:::; xn+m 2 R : ' defines the image of S under the projection n+m n π : R ! R ; (x1;:::; xn;:::; xn+m) 7! (x1;:::; xn): Disregarding algorithmic issues, real quantifier elimination thus simply says that projections of semialgebraic sets are again semialgebraic. Describing convex semialgebraic sets by LMIs To eliminate quantifiers in a formula, it suffices to eliminate one quantifier at a time (since 8x 2 R : ' is equivalent to :9x 2 R : :').