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Conic Optimization

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Conic Optimization CSCI 1951-G – Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34 This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of the figures are from S. Boyd, L. Vandenberge’s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/. 2 / 34 Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 3 / 34 Cones A set C is a cone if for every x C and θ 0, 2 ≥ θx C 2 2 Example: (x; x ); x R R f j j 2 g ⊂ Is this set convex? 4 / 34 Convex Cones A set C is a convex cone if, for every x1; x2 C and θ1; θ2 0, 26 2 2 Convex≥ sets θ1x1 + θ2x2 C: 2 Example: x1 x2 0 Figure 2.4 The pie slice shows all points of the form θ1x1 + θ2x2,where θ1, θ2 0. The apex of the slice (which corresponds to θ1 = θ2 = 0) is at ≥ 0; its edges (which correspond to θ1 = 0 or θ2 = 0) pass through the points x1 and x2. 5 / 34 0 0 Figure 2.5 The conic hulls (shown shaded) of the two sets of figure 2.3. Conic optimization Conic optimization problem in standard form: min cTx Ax = b x C 2 where C is a convex cone in finite-dimensional vector space X. Note: linear objective function, linear constraints. n n If X = R and C = R+, then ...we get an LP! Conic optimization is a unifying framework for linear programming, • second-order cone programming (SOCP), • semidefinite programming (SDP). • 6 / 34 Norm cones n−1 Let be any norm in R . k · k The norm cone associated to is the set k · k C = x = (x1; : : : ; xn): x1 (x2; : : : ; xn) f ≥ k kg It is a convex set. 7 / 34 3 2.2 Some important examplesSecond-order cone in R 31 The second-order cone is the norm cone for the Euclidean norm 2. k · k 1 0.5 t 0 1 1 0 0 x2 1 1 − − x1 3 2 2 1/2 Figure 2.10 Boundary of second-order cone in R , (x1,x2,t) (x1+x2) t . { | ≤ } What happens when we slice the second-order cone? It is (asI.e., the name when suggests) we take a convex the cone.intersection with a hyperplane? We obtain ellipsoidal sets. Example 2.3 The second-order cone is the norm cone for the Euclidean norm, i.e., 8 / 34 n+1 C = (x, t) R x 2 t { ∈ | ∥ ∥ ≤ } T x x I 0 x = 0,t 0 . ! t $ t 0 1 t ≤ ≥ % " # $ " # " − #" # $ $ The second-order cone is also known$ by several other names. It is calledthequadratic cone, since it is defined by a quadratic inequality. It is also called the Lorentz cone or ice-cream cone. Figure 2.10 shows the second-order cone in R3. 2.2.4 Polyhedra A polyhedron is defined as the solution set of a finite number of linear equalities and inequalities: = x aT x b ,j=1,...,m, cT x = d ,j=1,...,p . (2.5) P { | j ≤ j j j } A polyhedron is thus the intersection of a finite number of halfspaces and hyper- planes. Affine sets (e.g., subspaces, hyperplanes, lines), rays, line segments, and halfspaces are all polyhedra. It is easily shown that polyhedra are convex sets. A bounded polyhedron is sometimes called a polytope, but some authors use the opposite convention (i.e., polytope for any set of the form (2.5), and polyhedron Rewriting constraints Let’s rewrite C = x = (x1; : : : ; xn): x1 (x2; : : : ; xn) 2 f ≥ k k g as 2 2 2 x1 0; x x x 0 ≥ 1 − 2 − · · · n ≥ This is a combination of an linear and a quadratic constraints. Also: convex quadratic constraints can be expressed as second-order cone membership constraints. 9 / 34 Rewriting constraints adratic constraint: xT P x + 2qTx + γ 0 ≤ Assume P w.l.o.g. positive definite, so the constraint is ...convex. Also assume, for technical reasons, that qTP q γ 0. − ≥ Goal: rewrite the above constraint as a combination of linear and second-order cone membership constraints. 10 / 34 Rewriting constraints Because P is positive definitive, it has a Cholesky decomposition: invertible R s.t. P = RRT : 9 Rewrite the constraint as: (RTx)T(RTx) + 2qTx + γ 0 ≤ Let T T −1 y = (y1; : : : ; yn) = R x + R q The above is a bijection between x and y. We are going to rewrite the constraint as a constraint on y. 11 / 34 Rewriting constraints The constraint: (RTx)T(RTx) + 2qTx + γ 0 ≤ It holds yTy = (RTx)T(RTx) + 2qTx + qTP −1q Since there is a bijection between y and x, the constraint can be satisfied if and only if y s.t. y = RTx + R−1q; yTy qTP q γ 9 ≤ − 12 / 34 Rewriting constraints The constraint is equivalent to: y s.t. y = RTx + R−1q; yTy qTP q γ 9 ≤ − Lets denote with y0 the square root of the r.h.s. of the right inequality: T y0 = q P q γ R+ − 2 Consider the vector (y0;y 1;p : : : ; yn). y The right inequality then is | {z } n y2 yTy = y2 0 ≥ i i=1 X Taking the square root on both sides: n 2 y0 y = y 2 ≥ v i k k ui=1 uX This is the membership constraintt for the second-order cone in n+1. R 13 / 34 Rewriting constraints We rewrite the convex quadratic constraint xT P x + 2qTx + γ 0 ≤ as T T −1 (y1; : : : ; yn) = R x + R q T y0 = q P q γ R+ − 2 (y0; y1; : : : ; yn) C 2 p which is a combination of linear and second-order cone membership constraints. 14 / 34 Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 6. SDP and approximation algorithms 15 / 34 Benchmark-relative portfolio optimization Given a benchmark strategy xB (e.g., an index) develop a portfolio x that tracks xB, but adds value by beating it. I.e., we want a portfolio x with positive expected excess return: T µ (x xB) 0 − ≥ and specifically want to maximize the expected excess return. Challenge: balance expected excess return with its variance. 16 / 34 Tracking error and volatility constraints The (predicted) tracking error of the portfolio x is T TE(x) = (x xB) Σ(x xB) − − q It measure the variability of excess returns. In benchmark-relative portfolio optimization, we solve mean-variance optimization w.r.t. the expected excess return and tracking error: T max µ (x xB) − T 2 (x xB) Σ(x xB) T − − ≤ Ax = b 17 / 34 Comparison with mean-variance optimization We have seen MVO as: 1 T min x Σx δ 2 max µTx Σx µTx R or − 2 ≥ Ax = b Ax = b How do they dier from T max µ (x xB) − T 2 (x xB) Σ(x xB) T − − ≤ Ax = b The laer is not a standard quadratic program: it has a nonlinear constraint. 18 / 34 T max µ (x xB) − T 2 (x xB) Σ(x xB) T − − ≤ Ax = b The nonlinear constraint is ...convex quadratic We can rewrite it as a combination of linear and second-order cone membership, and solve the resulting convex conic problem. 19 / 34 Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 6. SDP and approximation algorithms 20 / 34 SemiDefinite Programming (SDP) The variables are the entries of a symmetric matrix in the cone of 2.3 Operations that preserve convexity 35 positive semidefinite matrices. 1 0.5 z 0 1 1 0 0.5 y 1 0 − x Figure 2.12 Boundary of positive semidefinite cone in S2. 21 / 34 x 0, y 0, xz y2. ≥ ≥ n ≥ n n The set S+ is a convex cone: if θ1, θ2 0 and A, B S+, then θ1A+θ2B S+. This can be seen directly from the definition≥ of positive∈ semidefiniteness: for∈ any x Rn, we have ∈ xT (θ A + θ B)x = θ xT Ax + θ xT Bx 0, 1 2 1 2 ≥ if A 0, B 0 and θ , θ 0. ≽ ≽ 1 2 ≥ Example 2.6 Positive semidefinite cone in S2. We have xy X = S2 x 0,z 0,xz y2. yz ∈ + ⇐⇒ ≥ ≥ ≥ ! " The boundary of this cone is shown in figure 2.12, plotted in R3 as (x, y, z). 2.3 Operations that preserve convexity In this section we describe some operations that preserve convexity of sets, or allow us to construct convex sets from others. These operations, together with the simple examples described in 2.2, form a calculus of convex sets that is useful for determining or establishing convexity§ of sets. Application: approximating covariance matrices Portfolio Optimization almost always requires covariance matrices. These are not directly available, but are estimated. Estimation of covariance matrices is a very challenging task, mathematically and computationally, because the matrices must satisfy various properties (e.g., symmetry, positive semidefiniteness). To be eicient, many estimation methods do not impose problem-dependent constraints. Typically, one is interested in finding the smallest distortion of the original estimate that satisfies the desired constraints; 22 / 34 Application: approximating covariance matrices Let Σ^ n be an estimate of a covariance matrix • 2 S Σ^ is symmetric ( n) but not positive semidefinite.
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