Basic model

Show that the following three convex problems are equivalent.
(a) The robust least-squares problem (Huber Regression)
min ⁡ ∑ i = 1 m ϕ ( a i T x − b i ) \min \sum_{i=1}^m\phi(a_i^Tx-b_i) mini=1∑mϕ(aiTx−bi)
with variable x ∈ R n x\in\mathbb{R}^n x∈Rn, where ϕ : R → R \phi:\mathbb{R}\to\mathbb{R} ϕ:R→R is defined as
ϕ ( u ) = { u 2 ∣ u ∣ ≤ M M ( 2 ∣ u ∣ − M ) ∣ u ∣ > M \phi(u)= \begin{cases} u^2 & |u|\leq M\\ M(2|u|-M)& |u|>M \end{cases} ϕ(u)={u2M(2∣u∣−M)∣u∣≤M∣u∣>M
(b) The least-squares problem with variable weights
min ⁡ ∑ i = 1 m ( a i T x − b i ) 2 / ( 1 + w i ) + M 2 1 T w s . t . w ⪰ 0 \min \sum_{i=1}^m (a_i^Tx-b_i)^2/(1+w_i)+M^2\mathbf{1}^Tw\\ s.t.\quad w\succeq 0 mini=1∑m(aiTx−bi)2/(1+wi)+M21Tws.t.w⪰0
quad_over_lin Sum of squares over linear. Z=quad_over_lin(X, Y) where X X X is a vector and Y Y Y is a scalar, is equal to sum(abs(X).^2)./Y if Y Y Y is positive, and ∞ \infty ∞ otherwise.

b.cvx code

%% cvx: least squares problem with variable weights
disp('Computing the solution of the least-squares problem with variable weights...');
cvx_beginvariable x2(n);variable w(m);minimize (sum(quad_over_lin(diag(A*x2-b), w'+1))+M^2*ones(1, m)*w);subject tow>=0;
cvx_end

(c) Quadratic program
min ⁡ ∑ i = 1 m ( u i 2 + 2 M v i ) s . t . { − u − v ⪯ A x − b ⪯ u + v 0 ⪯ u ⪯ M 1 \min \sum_{i=1}^m (u_i^2+2Mv_i)\\ s.t. \begin{cases} -u-v\preceq Ax-b\preceq u+v\\ 0\preceq u \preceq M\mathbf{1} \end{cases} mini=1∑m(ui2+2Mvi)s.t.{−u−v⪯Ax−b⪯u+v0⪯u⪯M1

c.cvx code

%% cvx: quadratic program
disp('Computing the solution of the quadratic program...');
cvx_beginvariable x3(n)variable u(m)variable v(m)minimize (sum(square(u)+2*M*v))A*x3-b<=u+v;A*x3-b>=-u-v;u>=0;u<=M;v>=0;
cvx_end

c.yalmip code

%% yalmip: quadratic program
x4 = sdpvar(n, 1);
u = sdpvar(m, 1);
v = sdpvar(m, 1);
F = [A*x4-b<=u+v, A*x4-b>=-u-v];
F = [F, u>=0, u<=M, v>=0];
obj = sum(u.^2+2*M*v);
optimize(F, obj);

Log-optimal investment strategy

We consider a portfolio problem with n n n assets held over N N N periods. At the begining of each period, we reinvest our total wealth, redistributing it over the n n n assets using a fixed, constant, allocation strategy x ∈ R n x\in\mathbb{R}^n x∈Rn, where x ⪰ 0 , 1 T x = 1 x\succeq 0, \mathbf{1}^Tx=1 x⪰0,1Tx=1. In other words, if W ( t − 1 ) W(t-1) W(t−1) is the wealth at the beginning of period t t t, then during period t t t we invest x i W ( t − 1 ) x_iW(t-1) xiW(t−1) in asset i i i. We denote by λ ( t ) \lambda(t) λ(t) the total return during period t t t, i.e., λ ( t ) = W ( t ) / W ( t − 1 ) \lambda(t)=W(t)/W(t-1) λ(t)=W(t)/W(t−1). At the end of N N N periods the wealth has been multiplied by the factor ∏ t = 1 N λ ( t ) \prod_{t=1}^N\lambda(t) ∏t=1Nλ(t). We call
1 N ∑ t = 1 N log ⁡ λ ( t ) \frac{1}{N}\sum_{t=1}^N\log\lambda(t) N1t=1∑Nlogλ(t)
the growth rate of the investment over the N N N periods. We are interested in determining an allocation strategy x x x that maximizing growth of the total wealth for large N N N.
A discrete stochastic model is employed to account for the uncertainty in the returns. We assume that during each period there are m m m possible scenarios, with probabilities π j , j = 1 , … , m \pi_j, j=1, \dots, m πj,j=1,…,m. In scenario j j j, the return for asset i i i over one period is given by p i j p_{ij} pij. Therefore, the return λ ( t ) \lambda(t) λ(t) of the portfolio during period t t t is a random variable, with m m m possible values p 1 T x , … , p m T x p_1^Tx, \dots, p_m^Tx p1Tx,…,pmTx, and distribution
π j = P ( λ ( t ) = p j T x ) , j = 1 , … , m \pi_j=\mathbb{P}(\lambda(t)=p_j^Tx), j=1,\dots, m πj=P(λ(t)=pjTx),j=1,…,m
We assume the same scenarios for each period, with identical independent distributions. Using the law of big numbers, we have
lim ⁡ N → ∞ 1 N ( W ( N ) W ( 0 ) ) = lim ⁡ N → ∞ 1 N ∑ t = 1 N log ⁡ λ ( t ) = ∑ j = 1 m π j log ⁡ ( p j T x ) \lim_{N\to\infty} \frac{1}{N}(\frac{W(N)}{W(0)})=\lim_{N\to\infty}\frac{1}{N}\sum_{t=1}^N\log\lambda(t)=\sum_{j=1}^m\pi_j\log(p_j^Tx) N→∞limN1(W(0)W(N))=N→∞limN1t=1∑Nlogλ(t)=j=1∑mπjlog(pjTx)
In other words, with investment strategy x x x, the long term growth rate is given by
R = ∑ j = 1 m π j log ⁡ ( p j T x ) R=\sum_{j=1}^m\pi_j\log(p_j^Tx) R=j=1∑mπjlog(pjTx)
The investment strategy x x x that maximizes this quantity is called the log-optimal investment strategy, and can be found by solving the optimization problem
max ⁡ ∑ j = 1 m π j log ⁡ ( p j T x ) s . t . { x ⪰ 0 1 T x = 1 \max \sum_{j=1}^m \pi_j\log(p_j^Tx)\\ s.t. \begin{cases} x\succeq 0\\ \mathbf{1}^Tx=1 \end{cases} maxj=1∑mπjlog(pjTx)s.t.{x⪰01Tx=1

cvx/yalmip code

%% Data
rng(729);
% P_{ij} is the return of asset i over one period in scenario j
P = [3.5000    1.1100    1.1100    1.0400    1.0100;0.5000    0.9700    0.9800    1.0500    1.0100;0.5000    0.9900    0.9900    0.9900    1.0100;0.5000    1.0500    1.0600    0.9900    1.0100;0.5000    1.1600    0.9900    1.0700    1.0100;0.5000    0.9900    0.9900    1.0600    1.0100;0.5000    0.9200    1.0800    0.9900    1.0100;0.5000    1.1300    1.1000    0.9900    1.0100;0.5000    0.9300    0.9500    1.0400    1.0100;3.5000    0.9900    0.9700    0.9800    1.0100];% n assets and m scenarios[m, n] = size(P);v = abs(randn(m, 1));Pi = v/sum(v);x_eq = ones(n, 1)/n;%% cvx: find the log-optimal investment policycvx_beginvariable x_opt(n);maximize sum(Pi.*log(P*x_opt))subject tosum(x_opt)==1;x_opt>=0;cvx_end%% yalmipx_yal = sdpvar(n, 1);F = [sum(x_yal)==1, x_yal>=0];obj = sum(Pi.*log(P*x_yal));optimize(F, -obj);x_yal = value(x_yal);%% long-term growth ratesR_opt = sum(Pi.*log(P*x_opt));R_eq = sum(Pi.*log(P*x_eq));disp('The long term growth rate of the log-optimal strategy is:');disp(R_opt);disp('The long term growth rate of the uniform strategy is:');disp(R_eq);%% random event sequencesN = 10;T = 200;w_opt = zeros(1+T, N);w_eq = zeros(1+T, N);for i=1:Nevents = ceil(rand(1, T)*m);P_event = P(events, :);w_opt(:, i) = [1; cumprod(P_event*x_opt)];w_eq(:, i) =  [1; cumprod(P_event*x_eq)];endfigure;semilogy(w_opt, 'b');hold onsemilogy(w_eq, 'r');axis tightxlabel('time')ylabel('wealth')

Reference

Convex Optimization S.Boyd. Page 233

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【CVX】Equivalent convex problems (Huber) Log-optimal investment strategy

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