• Tarski-Seidenberg and quantifier elimination
• Boolean minimization and the S-procedure
The basic (closed) semialgebraic set defined by polynomials f1, . . . , fm is
x ∈ Rn | fi(x) ≥ 0 for all i = 1, . . . , m
• The cone of positive semidefinite matrices
• Feasible set of an SDP; polyhedra and spectrahedra
• If S1, S2 are basic closed semialgebraic sets, then so is S1 ∩ S2; i.e.,
Given the basic semialgebraic sets, we may generate other sets by set the-oretic operations; unions, intersections and complements.
A set generated by a finite sequence of these operations on basic semial-gebraic sets is called a semialgebraic set.
is semialgebraic, where ∗ denotes <, ≤, =, =.
• In particular every real variety is semialgebraic.
• We can also generate the semialgebraic sets via Boolean logical oper-
ations applied to polynomial equations and inequalities
Every semialgebraic set may be represented as either
x ∈ Rn | fi(x) > 0, hj(x) = 0 for all i = 1, . . . , m, j = 1, . . . , p
• in R, a finite union of points and open intervals
Every closed semialgebraic set is a finite union of basic closed semialgebraicsets; i.e., sets of the form
x ∈ Rn | fi(x) ≥ 0 for all i = 1, . . . , m
• If S1, S2 are semialgebraic, so is S1 ∪ S2 and S1 ∩ S2
• The projection of a semialgebraic set is semialgebraic
• The closure and interior of a semialgebraic sets are both semialgebraic
Some sets are not semialgebraic; for example
• the infinite staircase (x, y) ∈ R2 | y = x
Tarski-Seidenberg and Quantifier Elimination
Tarski-Seidenberg theorem: if S ⊂ Rn+p is semialgebraic, then so are
i.e., quantifiers do not add any expressive power
Cylindrical algebraic decomposition (CAD) may be used to compute thesemialgebraic set resulting from quantifier elimination
Suppose S is a semialgebraic set; we’d like to solve the feasibility problem
More specifically, suppose we have a semialgebraic set represented by poly-nomial inequalities and equations
x ∈ Rn | fi(x) ≥ 0, hj(x) = 0 for all i = 1, . . . , m, j = 1, . . . , p
• Important, non-trivial result: the feasibility problem is decidable. • But NP-hard (even for a single polynomial, as we have seen)• We would like to certify infeasibility
• The Nullstellensatz: a necessary and sufficient condition for feasibility
• Valid inequalities: a sufficient condition for infeasibility of real basic
• Linear Programming: necessary and sufficient conditions via duality
for real linear equations and inequalities
We’d like a method to construct certificates for
If we can test feasibility of real equations then we can also test feasibilityof real inequalities and inequations, because
• inequalities: there exists x ∈ R such that f(x) ≥ 0 if and only if
there exists (x, y) ∈ R2 such that f(x) = y2
• strict inequalities: there exists x such that f(x) > 0 if and only if
there exists (x, y) ∈ R2 such that y2f(x) = 1
• inequations: there exists x such that f(x) = 0 if and only if
there exists (x, y) ∈ R2 such that yf(x) = 1
The underlying theory for real polynomials called real algebraic geometry
The real variety defined by polynomials h1, . . . , hm ∈ R[x1, . . . , xn] is
VR{h1, . . . , hm} = x ∈ Rn | hi(x) = 0 for all i = 1, . . . , m
We’d like to solve the feasibility problem; is VR{h1, . . . , hm} = ∅?
• Every polynomial in ideal{h1, . . . , hm} vanishes on the feasible set.
• But this condition is not necessary over the reals
Recall Σ is the cone of polynomials representable as sums of squares.
Suppose h1, . . . , hm ∈ R[x1, . . . , xn].
Equivalently, there is no x ∈ Rn such that
if and only if there exists t1, . . . , tm ∈ R[x1, . . . , xn] and s ∈ Σ such that
Suppose h(x) = x2 + 1. Then clearly VR{h} = ∅
We saw earlier that the complex Nullstellensatz cannot be used to proveemptyness of VR{h}
and so the real Nullstellensatz implies VR{h} = ∅.
The polynomial equation −1 = s + th gives a certificate of infeasibility.
We now turn to feasibility for basic semialgebraic sets, with primal problem
• every polynomial in cone{f1, . . . , fm} is nonnegative on S
• every polynomial in ideal{h1, . . . , hp} is zero on S
−1 ∈ cone{f1, . . . , fm} + ideal{h1, . . . , hm}
(x, y) ∈ R2 | f(x, y) ≥ 0, h(x, y) = 0
By the P-satz, the primal is infeasible if and only if there exist polynomialss1, s2 ∈ Σ and t ∈ R[x, y] such that
Explicit Formulation of the Positivstellensatz
Do there exist ti ∈ R[x1, . . . , xn] and si, rij, . . . ∈ Σ such that
Do there exist ti ∈ R[x1, . . . , xn] and si, rij, . . . ∈ Σ such that
• This is a convex feasibility problem in ti, si, rij, . . . • To solve it, we need to choose a subset of the cone to search; i.e.,
the maximum degree of the above polynomial; then the problem is asemidefinite program
• This gives a hierarchy of syntactically verifiable certificates
• The validity of a certificate may be easily checked; e.g., linear algebra,
• Unless NP=co-NP, the certificates cannot always be polynomially sized.
The primal problem; does there exist x ∈ Rn such that
Let fi(x) = aTi x + bi, hi(x) = cTi x + di. Then this system is infeasible if
−1 ∈ cone{f1, . . . , fm} + ideal{h1, . . . , hp}
Searching over linear combinations, the primal is infeasible if there existλ ≥ 0 and µ such that
Equating coefficients, this is equivalent to
λT A + µT C = 0 λT b + µT d = −1 λ ≥ 0
• Interesting connections with logic, proof systems, etc. • Failure to prove infeasibility (may) provide points in the set. • Tons of applications:
optimization, copositivity, dynamical systems, quantum mechanics.
Many known methods can be interpreted as fragments of P-satz refutations.
• LP duality: linear inequalities, constant multipliers. • S-procedure: quadratic inequalities, constant multipliers• Standard SDP relaxations for QP. • The linear representations approach for functions f strictly positive on
• Losslessness: when can we restrict a priori the class of certificates?• Some cases are known; e.g., additional conditions such as linearity, per-
fect graphs, compactness, finite dimensionality, etc, can ensure specifica priori properties.
which holds if and only if there exists a diagonal Λ such that Q
γ = trace Λ − ε. The corresponding optimization problem is
The primal problem; does there exist x ∈ Rn such that
We have a P-satz refutation if there exists λ1, λ2 ≥ 0, µ ∈ R and S
−1 = xT Sx + λ1xT F1x + λ2xT F2x + µ(1 − xT x)
which holds if and only if there exist λ1, λ2 ≥ 0 such that
Subject to an additional mild constraint qualification, this condition is alsonecessary for infeasibility.
What algebraic properties of the polynomial system yield efficient compu-tation?
• Sparseness: few nonzero coefficients.
• Newton polytopes techniques• Complexity does not depend on the degree
• Symmetries: invariance under a transformation group
• Frequent in practice. Enabling factor in applications. • Can reflect underlying physical symmetries, or modelling choices. • SOS on invariant rings• Representation theory and invariant-theoretic techniques.
• Ideal structure: Equality constraints.
• SOS on quotient rings• Compute in the coordinate ring. Quotient bases (Groebner)
• Structured singular value µ and related problems: provides better up-
• µ is a measure of robustness: how big can a structured perturbation
• A standard semidefinite relaxation: the µ upper bound.
• Morton and Doyle’s counterexample with four scalar blocks. • Exact value: approx. 0.8723• Standard µ upper bound: 1• New bound: 0.895
• The set of copositive matrices is a convex closed cone, but.
• Checking copositivity is coNP-complete
• Very important in QP. Characterization of local solutions.
• The P-satz gives a family of computable SDP conditions, via:
Ono’s inequality: For an acute triangle,
(4K)6 ≥ 27 · (a2 + b2 − c2)2 · (b2 + c2 − a2)2 · (c2 + a2 − b2)2
where K and a, b, c are the area and lengths of the edges.
s(x, y, z) = (x4 + x2y2 − 2y4 − 2x2z2 + y2z2 + z4)2 + 15 · (x − z)2(x + z)2(z2 + x2 − y2)2.
(4K)6 − 27 · t21 · t22 · t23 = s(a, b, c) · t1 · t2 + s(c, a, b) · t1 · t3 + s(b, c, a) · t2 · t3

CANINE BRAIN TUMORS What are brain tumors? Primary brain tumors are relatively uncommon in dogs. Brain tumors include a broad spectrum of tumor types. Gliomas and meningiomas are the most common type in dogs. Brain tumors occur most frequently in older dogs (over 5 years) no sex predilection. Boxers have an increased incidence of meningiomas. The most common secondary tumors in dogs

DEMOGRAPHICS OF GUANTANAMO BAY PRISON* I. THE HISTORY OF U.S. CONTROL OF GUANTANAMO BAY, CUBA The United States joined the Cubans in their war against their colonial ruler, Spain, in February 1898 after the U.S. warship, the U.S.S. Maine, was blown up by the Spanish in Havana harbour. In June 1898, the United States military forces captured Guantanamo Bay, located in the south-eastern co