CSCI 1951-G Optimization Methods in Finance Part 10: Conic - - PowerPoint PPT Presentation

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, θx ∈ C Example: {(x, |x|), x ∈ R} ⊂ R2 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, θ1x1 + θ2x2 ∈ C . Example:

x1 x2

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

Conic optimization

Conic optimization problem in standard form: min cTx Ax = b x ∈ C where C is a convex cone in finite-dimensional vector space X. Note: linear objective function, linear constraints. If X = Rn and C = Rn

+, 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

Let · be any norm in Rn−1. The norm cone associated to · is the set C = {x = (x1, . . . , xn) : x1 ≥ (x2, . . . , xn)} It is a convex set.

7 / 34

Second-order cone in R3

The second-order cone is the norm cone for the Euclidean norm · 2.

x1 x2 t −1 1 −1 1 0.5 1

Figure 2.10 Boundary of second-order cone in R3, {(x1, x2, t) | (x2

1+x2 2)1/2 ≤

t}.

What happens when we slice the second-order cone? I.e., when we take the intersection with a hyperplane? We obtain ellipsoidal sets.

8 / 34

Rewriting constraints

Let’s rewrite C = {x = (x1, . . . , xn) : x1 ≥ (x2, . . . , xn)2} as x1 ≥ 0, x2

1 − x2 2 − · · · x2 n ≥ 0

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

Qadratic constraint: xT Px + 2qTx + γ ≤ 0 Assume P w.l.o.g. positive definite, so the constraint is ...convex. Also assume, for technical reasons, that qTPq − 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 . Rewrite the constraint as: (RTx)T(RTx) + 2qTx + γ ≤ 0 Let y = (y1, . . . , yn)T = RTx + R−1q 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 ≤ qTPq − γ

12 / 34

Rewriting constraints

The constraint is equivalent to: ∃y s.t. y = RTx + R−1q, yTy ≤ qTPq − γ Lets denote with y0 the square root of the r.h.s. of the right inequality: y0 =

• qTPq − γ ∈ R+

Consider the vector (y0, y1, . . . , yn

• y

). The right inequality then is y2

0 ≥ yTy = n

• i=1

y2

i

Taking the square root on both sides: y0 ≥

• n
• i=1

y2

i = y2

This is the membership constraint for the second-order cone in Rn+1.

13 / 34

Rewriting constraints

We rewrite the convex quadratic constraint xT Px + 2qTx + γ ≤ 0 as (y1, . . . , yn)T = RTx + R−1q y0 =

• qTPq − γ ∈ R+

(y0, y1, . . . , yn) ∈ C 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: µT(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 TE(x) =

• (x − xB)TΣ(x − xB)

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: max µT(x − xB) (x − xB)TΣ(x − xB) ≤ T 2 Ax = b

17 / 34

Comparison with mean-variance optimization

We have seen MVO as: min 1 2xTΣx µTx ≥ R Ax = b

• r

max µTx − δ 2Σx Ax = b How do they differ from max µT(x − xB) (x − xB)TΣ(x − xB) ≤ T 2 Ax = b The later is not a standard quadratic program: it has a nonlinear constraint.

18 / 34

max µT(x − xB) (x − xB)TΣ(x − xB) ≤ T 2 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 positive semidefinite matrices.

x y z 0.5 1 −1 1 0.5 1

Figure 2.12 Boundary of positive semidefinite cone in S2.

21 / 34

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 efficient, many estimation methods do not impose problem-dependent constraints. Typically, one is interested in finding the smallest distortion of the

• riginal estimate that satisfies the desired constraints;

22 / 34

Application: approximating covariance matrices

• Let ˆ

Σ ∈ Sn be an estimate of a covariance matrix

• ˆ

Σ is symmetric (∈ Sn) but not positive semidefinite. Goal: find the positive semidefinite matrix that is closest to ˆ Σ w.r.t. the Frobenius norm: dF (Σ, ˆ Σ) =

• i,j

(Σij − ˆ Σij)2 Formally: nearest covariance matrix problem: min

Σ

dF (Σ, ˆ Σ) Σ ∈ Cn

s

where Cn

s is the cone of n × n symmetric and positive semidefinite

matrices.

23 / 34

Application: approximating covariance matrices

min

Σ

dF (Σ, ˆ Σ) Σ ∈ Cn

s

Introduce a dummy variable t and rewrite the problem as min t dF (Σ, ˆ Σ) ≤ t Σ ∈ Cn

s

The first constraint can be writen as a second-order cone constraint, so the problem is transformed into a conic optimization problem.

24 / 34

Application: approximating covariance matrices

Variation of the problem with additional linear constraints: Let E ⊆ {(i, j) : 1 ≤ i ≤ n} Let (ℓij, uij), for (i, j) ∈ E be lower/upper bounds to impose to the entries. We want to solve: min

Σ

dF (Σ, ˆ Σ) ℓij < Σij < uij, ∀(i, j) ∈ E Σ ∈ Cn

s

25 / 34

Application: approximating covariance matrices

For example, let ˆ Σ be an estimation of a correlation matrix. Correlation matrix have all diagonal entries equal to 1. We want to solve the nearest correlation matrix problem. We choose E = {(i, i), 1 ≤ i ≤ n}, ℓi = 1 = ui, 1 ≤ i ≤ n

26 / 34

Application: approximating covariance matrices

Many other variants are possible:

• Force some entries of ˆ

Σ to remain the same in Σ;

• Weight the changes to different entries differently, because we

trust some more than other;

• Impose lower bounds to the minimum eigenvalue of Σ, to reduce

instability; All of these can be easily solved with SDP sofware.

27 / 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

28 / 34

A different point of view to SDP

A n × n matrix A is positive semidefinite if there are vectors xi, . . . xj such that Aij = xT

i xj.

We can then write a semidefinite program as a program involving

• nly linear combinations of the inner products of the vectors

xi, . . . xj: min

• i,j∈[n]

cijxT

i xj

• i,j∈[n]

aijkxT

i xj ≤ bj, ∀k

This form is particularly useful to develop approximation algorithms.

29 / 34

The MaxCut problem

Given a graph G = (V, E), output a 2-partition of V so as to maximize the number of edges crossing from one side to the other. Integer quadratic program: max

• (i,j)∈E

1 − vivj 2 vi ∈ {−1, 1}, 1 ≤ i ≤ n The decision version of the problem is NP-complete.

30 / 34

The MaxCut problem

Steps for an approximation algorithm for MaxCut: 1 Relax the original problem to an SDP; 2 Solve the SDP; 3 Round the SDP solution to obtain an integer solution to the

• riginal problem.

31 / 34

The MaxCut problem

Integer quadratic program: max

• (i,j)∈E

1 − vivj 2 vi ∈ {−1, 1}, 1 ≤ i ≤ n SDP Relaxation: max

• (i,j)∈E

1 − vT

i vj

2 vi2

2 ≤ 1, 1 ≤ i ≤ n

vi ∈ Rn It is a relaxation: the optimal obj. value will be larger than the one for the original problem.

32 / 34

The MaxCut problem

max

• (i,j)∈E

1 − vT

i vj

2 vi2

2 ≤ 1, 1 ≤ i ≤ n

vi ∈ Rn The optimal solution is a set of unit vectors in Rn To obtain a solution for the original problem, we need to round this solution and assign each vector to one value in {−1, 1}. Goemans and Willamson 1995: choose a random hyperplane that goes through the origin, and split the vectors depending on the side

• f the hyperplane.

Approximation ratio: 0.87856 - ε (essentially optimal)

33 / 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

34 / 34