8. Geometric problems
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1 8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 1
2 Minimum volume ellipsoid around a set Löwner-John ellipsoid of a set C: minimum volume ellipsoid E s.t. C E parametrize E as E = {v Av +b 2 1}; w.l.o.g. assume A S n ++ vole is proportional to deta 1 ; to compute minimum volume ellipsoid, minimize (over A, b) logdeta 1 subject to sup v C Av +b 2 1 convex, but evaluating the constraint can be hard (for general C) finite set C = {x 1,...,x m }: minimize (over A, b) logdeta 1 subject to Ax i +b 2 1, i = 1,...,m also gives Löwner-John ellipsoid for polyhedron conv{x 1,...,x m } Geometric problems 8 2
3 Maximum volume inscribed ellipsoid maximum volume ellipsoid E inside a convex set C R n parametrize E as E = {Bu+d u 2 1}; w.l.o.g. assume B S n ++ vole is proportional to detb; can compute E by solving maximize log det B subject to sup u 2 1I C (Bu+d) 0 (where I C (x) = 0 for x C and I C (x) = for x C) convex, but evaluating the constraint can be hard (for general C) polyhedron {x a T i x b i, i = 1,...,m}: maximize log det B subject to Ba i 2 +a T i d b i, i = 1,...,m (constraint follows from sup u 2 1a T i (Bu+d) = Ba i 2 +a T i d) Geometric problems 8 3
4 Efficiency of ellipsoidal approximations C R n convex, bounded, with nonempty interior Löwner-John ellipsoid, shrunk by a factor n, lies inside C maximum volume inscribed ellipsoid, expanded by a factor n, covers C example (for two polyhedra in R 2 ) factor n can be improved to n if C is symmetric Geometric problems 8 4
5 Centering some possible definitions of center of a convex set C: center of largest inscribed ball ( Chebyshev center ) for polyhedron, can be computed via linear programming (page 4 19) center of maximum volume inscribed ellipsoid (page 8 3) x cheb x mve MVE center is invariant under affine coordinate transformations Geometric problems 8 5
6 Analytic center of a set of inequalities the analytic center of set of convex inequalities and linear equations f i (x) 0, i = 1,...,m, Fx = g is defined as the optimal point of minimize m i=1 log( f i(x)) subject to Fx = g more easily computed than MVE or Chebyshev center (see later) not just a property of the feasible set: two sets of inequalities can describe the same set, but have different analytic centers Geometric problems 8 6
7 analytic center of linear inequalities a T i x b i, i = 1,...,m x ac is minimizer of φ(x) = m log(b i a T i x) i=1 x ac inner and outer ellipsoids from analytic center: E inner {x a T i x b i, i = 1,...,m} E outer where E inner = {x (x x ac ) T 2 φ(x ac )(x x ac ) 1} E outer = {x (x x ac ) T 2 φ(x ac )(x x ac ) m(m 1)} Geometric problems 8 7
8 Linear discrimination separate two sets of points {x 1,...,x N }, {y 1,...,y M } by a hyperplane: a T x i +b > 0, i = 1,...,N, a T y i +b < 0, i = 1,...,M homogeneous in a, b, hence equivalent to a T x i +b 1, i = 1,...,N, a T y i +b 1, i = 1,...,M a set of linear inequalities in a, b Geometric problems 8 8
9 Robust linear discrimination (Euclidean) distance between hyperplanes H 1 = {z a T z +b = 1} H 2 = {z a T z +b = 1} is dist(h 1,H 2 ) = 2/ a 2 to separate two sets of points by maximum margin, minimize (1/2) a 2 subject to a T x i +b 1, i = 1,...,N a T y i +b 1, i = 1,...,M (1) (after squaring objective) a QP in a, b Geometric problems 8 9
10 Lagrange dual of maximum margin separation problem (1) maximize 1 T λ+1 T µ subject to 2 N i=1 λ ix i M i=1 µ 2 iy i 1 1 T λ = 1 T µ, λ 0, µ 0 (2) from duality, optimal value is inverse of maximum margin of separation interpretation change variables to θ i = λ i /1 T λ, γ i = µ i /1 T µ, t = 1/(1 T λ+1 T µ) invert objective to minimize 1/(1 T λ+1 T µ) = t minimize t subject to N i=1 θ ix i M i=1 γ 2 iy i t θ 0, 1 T θ = 1, γ 0, 1 T γ = 1 optimal value is distance between convex hulls Geometric problems 8 10
11 Approximate linear separation of non-separable sets an LP in a, b, u, v minimize 1 T u+1 T v subject to a T x i +b 1 u i, i = 1,...,N a T y i +b 1+v i, i = 1,...,M u 0, v 0 at optimum, u i = max{0,1 a T x i b}, v i = max{0,1+a T y i +b} can be interpreted as a heuristic for minimizing #misclassified points Geometric problems 8 11
12 Support vector classifier minimize a 2 +γ(1 T u+1 T v) subject to a T x i +b 1 u i, i = 1,...,N a T y i +b 1+v i, i = 1,...,M u 0, v 0 produces point on trade-off curve between inverse of margin 2/ a 2 and classification error, measured by total slack 1 T u+1 T v same example as previous page, with γ = 0.1: Geometric problems 8 12
13 Nonlinear discrimination separate two sets of points by a nonlinear function: f(x i ) > 0, i = 1,...,N, f(y i ) < 0, i = 1,...,M choose a linearly parametrized family of functions f(z) = θ T F(z) F = (F 1,...,F k ) : R n R k are basis functions solve a set of linear inequalities in θ: θ T F(x i ) 1, i = 1,...,N, θ T F(y i ) 1, i = 1,...,M Geometric problems 8 13
14 quadratic discrimination: f(z) = z T Pz +q T z +r x T i Px i +q T x i +r 1, y T i Py i +q T y i +r 1 can add additional constraints (e.g., P I to separate by an ellipsoid) polynomial discrimination: F(z) are all monomials up to a given degree separation by ellipsoid separation by 4th degree polynomial Geometric problems 8 14
15 Placement and facility location N points with coordinates x i R 2 (or R 3 ) some positions x i are given; the other x i s are variables for each pair of points, a cost function f ij (x i,x j ) placement problem minimize i j f ij(x i,x j ) variables are positions of free points interpretations points represent plants or warehouses; f ij is transportation cost between facilities i and j points represent cells on an IC; f ij represents wirelength Geometric problems 8 15
16 example: minimize (i,j) A h( x i x j 2 ), with 6 free points, 27 links optimal placement for h(z) = z, h(z) = z 2, h(z) = z histograms of connection lengths x i x j Geometric problems 8 16
8. Geometric problems
8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 Minimum volume ellipsoid around a set Löwner-John ellipsoid
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