i have a problem that i'd like to solve in R.
I see that i can use the function lsei in the package limSolve to minimise a system of linear equations written Ax=b in the matrix form, subject to equality constraints Ex=f and the inequality constraints Gx>=h.
However, rather than a system of linear equations, i now have a system of quadratic equations that can be written t(x)Ax=b.
I see there's the package quadprog for the quadratic case, but it doesn't seem to allow for a set of quadratic equations, just the one equation.
Does anyone know what i could use to minimise a system of quadratic equations under both an equality and an inequality constraint?
Here's my example. I'm trying to combine 3 probabilities P(A), P(B), P(C) - this creates 7 segments v1 to v7, where v1 is P(A solus) etc... v4 is P((A AND B) NOT C) etc.. and v7 is P(A AND B AND C).
The function i'm trying to minimise is:
obj.fc<-function(x){
f<-rep(NA,4)
v1<-x[1]
v2<-x[2]
v3<-x[3]
v4<-x[4]
v5<-x[5]
v6<-x[6]
v7<-x[7]
f[1]<-(v4+v7)*(1-(v1+v2+v4+v5+v6+v7))-2*(v1+v6)*(v2+v5)
f[2]<-(v5+v7)*(1-(v2+v3+v4+v5+v6+v7))-13*(v2+v4)*(v3+v6)
f[3]<-(v6+v7)*(1-(v1+v3+v4+v5+v6+v7))-11*(v1+v4)*(v3+v5)
f[4]<-(v4+v5+v6)*(1-(v1+v2+v3+v4+v5+v6+v7))-4*(v1+v2+v3)*v7
return(f)
}
My equality constraints are:
v1+v4+v6+v7=0.14
v2+v4+v5+v7=0.01
v3+v5+v6+v7=0.08
And my inequality constraints are that the Vi have to be between 0 and 1 and their sum can't exceed 1.
You can represent each of your equality constraints as two inequality constraints, e.g.
Ax = b <=> Ax <= b, and
Ax >= b
Note however that quadprog only allows solving "quadratic programs" in the sense of a quadratic objective function with linear constraints. From quadprog documentation:
This routine implements the dual method of Goldfarb and Idnani (1982,
1983) for solving quadratic programming problems of the form min(−dT b
+ 1/2bT Db) with the constraints AT b >= b0.
So in your case, you should probably look at another package. I would suggest e.g. [NlcOptim][2], or, find the most appropriate solver for you from here:
https://cran.r-project.org/web/views/Optimization.html
Related
I've been able to minimize a non-linear objective with a linear constraint using quadprog, however, I haven't been able to do it the other way around...
require(quadprog)
min_var <- function(Obj,Rentabilidades,var_covar){
b <- c(Obj,1)
Betha <- var_covar
A <- t(matrix(rbind(Rentabilidades,c(1,1)),nrow=2))
Gamma <- matrix(0,nrow=2)
solve.QP(Betha,Gamma,A,b,2)
}
Now I want to maximize what previously was the constraint taking as a new constraint the former objective. Regrettably, solve.QP() only supports linear constraints. Does anyone know a package similar to quadprog that might help me?
A standard portfolio optimization model looks like:
min sum((i,j), x(i)*Q(i,j)*x(j))
sum(i,x(i)) = 1
sum(i,r(i)*x(i)) >= R
x(i) >= 0
This is a Quadratic Programming model, and can be solved with standard QP solvers.
If you turn this around (maximize return subject to a risk constraint), you can write:
max sum(i,r(i)*x(i))
sum((i,j), x(i)*Q(i,j)*x(j)) <= V
sum(i,x(i)) = 1
x(i) >= 0
This is now a Quadratically Constrained problem. Luckily this is convex so you can use solvers like Cplex, Gurobi, or Mosek to solve them (they have R interfaces). An open source candidate could be a solver like ECOSolveR, or even better a framework like cxvr.
I am trying to solve the following inequality constraint:
Given time-series data for N stocks, I am trying to construct a portfolio weight vector to minimize the variance of the returns.
the objective function:
min w^{T}\sum w
s.t. e_{n}^{T}w=1
\left \| w \right \|\leq C
where w is the vector of weights, \sum is the covariance matrix, e_{n}^{T} is a vector of ones, C is a constant. Where the second constraint (\left \| w \right \|) is an inequality constraint (2-norm of the weights).
I tried using the nloptr() function but it gives me an error: Incorrect algorithm supplied. I'm not sure how to select the correct algorithm and I'm also not sure if this is the right method of solving this inequality constraint.
I am also open to using other functions as long as they solve this constraint.
Here is my attempted solution:
data <- replicate(4,rnorm(100))
N <- 4
fn<-function(x) {cov.Rt<-cov(data); return(as.numeric(t(x) %*%cov.Rt%*%x))}
eqn<-function(x){one.vec<-matrix(1,ncol=N,nrow=1); return(-1+as.numeric(one.vec%*%x))}
C <- 1.5
ineq<-function(x){
z1<- t(x) %*% x
return(as.numeric(z1-C))
}
uh <-rep(C^2,N)
lb<- rep(0,N)
x0 <- rep(1,N)
local_opts <- list("algorithm"="NLOPT_LN_AUGLAG,",xtol_rel=1.0e-7)
opts <- list("algorithm"="NLOPT_LN_AUGLAG,",
"xtol_rel"=1.0e-8,local_opts=local_opts)
sol1<-nloptr(x0,eval_f=fn,eval_g_eq=eqn, eval_g_ineq=ineq,ub=uh,lb=lb,opts=opts)
This looks like a simple QP (Quadratic Programming) problem. It may be easier to use a QP solver instead of a general purpose NLP (NonLinear Programming) solver (no need for derivatives, functions etc.). R has a QP solver called quadprog. It is not totally trivial to setup a problem for quadprog, but here is a very similar portfolio example with complete R code to show how to solve this. It has the same objective (minimize risk), the same budget constraint and the lower and upper-bounds. The example just has an extra constraint that specifies a minimum required portfolio return.
Actually I misread the question: the second constraint is ||x|| <= C. I think we can express the whole model as:
This actually looks like a convex model. I could solve it with "big" solvers like Cplex,Gurobi and Mosek. These solvers support convex Quadratically Constrained problems. I also believe this can be formulated as a cone programming problem, opening up more possibilities.
Here is an example where I use package cccp in R. cccp stands for
Cone Constrained Convex Problems and is a port of CVXOPT.
The 2-norm of weights doesn't make sense. It has to be the 1-norm. This is essentially a constraint on the leverage of the portfolio. 1-norm(w) <= 1.6 implies that the portfolio is at most 130/30 (Sorry for using finance language here). You want to read about quadratic cones though. w'COV w = w'L'Lw (Cholesky decomp) and hence w'Cov w = 2-Norm (Lw)^2. Hence you can introduce the linear constraint y - Lw = 0 and t >= 2-Norm(Lw) [This defines a quadratic cone). Now you minimize t. The 1-norm can also be replaced by cones as abs(x_i) = sqrt(x_i^2) = 2-norm(x_i). So introduce a quadratic cone for each element of the vector x.
I am trying to solve the following inequality constraint:
Given time-series data for N stocks, I am trying to construct a portfolio weight vector to minimize the variance of the returns.
the objective function:
min w^{T}\sum w
s.t. e_{n}^{T}w=1
\left \| w \right \|\leq C
where w is the vector of weights, \sum is the covariance matrix, e_{n}^{T} is a vector of ones, C is a constant. Where the second constraint (\left \| w \right \|) is an inequality constraint (2-norm of the weights).
I tried using the nloptr() function but it gives me an error: Incorrect algorithm supplied. I'm not sure how to select the correct algorithm and I'm also not sure if this is the right method of solving this inequality constraint.
I am also open to using other functions as long as they solve this constraint.
Here is my attempted solution:
data <- replicate(4,rnorm(100))
N <- 4
fn<-function(x) {cov.Rt<-cov(data); return(as.numeric(t(x) %*%cov.Rt%*%x))}
eqn<-function(x){one.vec<-matrix(1,ncol=N,nrow=1); return(-1+as.numeric(one.vec%*%x))}
C <- 1.5
ineq<-function(x){
z1<- t(x) %*% x
return(as.numeric(z1-C))
}
uh <-rep(C^2,N)
lb<- rep(0,N)
x0 <- rep(1,N)
local_opts <- list("algorithm"="NLOPT_LN_AUGLAG,",xtol_rel=1.0e-7)
opts <- list("algorithm"="NLOPT_LN_AUGLAG,",
"xtol_rel"=1.0e-8,local_opts=local_opts)
sol1<-nloptr(x0,eval_f=fn,eval_g_eq=eqn, eval_g_ineq=ineq,ub=uh,lb=lb,opts=opts)
This looks like a simple QP (Quadratic Programming) problem. It may be easier to use a QP solver instead of a general purpose NLP (NonLinear Programming) solver (no need for derivatives, functions etc.). R has a QP solver called quadprog. It is not totally trivial to setup a problem for quadprog, but here is a very similar portfolio example with complete R code to show how to solve this. It has the same objective (minimize risk), the same budget constraint and the lower and upper-bounds. The example just has an extra constraint that specifies a minimum required portfolio return.
Actually I misread the question: the second constraint is ||x|| <= C. I think we can express the whole model as:
This actually looks like a convex model. I could solve it with "big" solvers like Cplex,Gurobi and Mosek. These solvers support convex Quadratically Constrained problems. I also believe this can be formulated as a cone programming problem, opening up more possibilities.
Here is an example where I use package cccp in R. cccp stands for
Cone Constrained Convex Problems and is a port of CVXOPT.
The 2-norm of weights doesn't make sense. It has to be the 1-norm. This is essentially a constraint on the leverage of the portfolio. 1-norm(w) <= 1.6 implies that the portfolio is at most 130/30 (Sorry for using finance language here). You want to read about quadratic cones though. w'COV w = w'L'Lw (Cholesky decomp) and hence w'Cov w = 2-Norm (Lw)^2. Hence you can introduce the linear constraint y - Lw = 0 and t >= 2-Norm(Lw) [This defines a quadratic cone). Now you minimize t. The 1-norm can also be replaced by cones as abs(x_i) = sqrt(x_i^2) = 2-norm(x_i). So introduce a quadratic cone for each element of the vector x.
My problem is one which should be quite common in statistical inference:
min{(P - k)'S(P - k)} subject to k >= 0
So my choice variable is k, a 3x1 vector. The 3x1 vector P and 3x3 matrix S are known. Is it possible to reformulate this problem so I can use R's solve.QP quadratic programming solver? This solver requires the problem to be in the form
min{-d'b + 0.5 b' D b} subject to A'b >= b_0.
So here the choice vector is is b. Is there a way I can make my problem fit into solve.QP? Thanks so much for any help.
I face the following problem in a cryptographical application: I have given a set of linear congruences
a[1]*x[1]+a[2]*x[2]+a[3]*x[3] == d[1] (mod p)
b[1]*x[1]+b[2]*x[2]+b[3]*x[3] == d[2] (mod p)
c[1]*x[1]+c[2]*x[2]+c[3]*x[3] == d[3] (mod p)
Here, x is unknown an a,b,c,d are given
The system is most likely underdetermined, so I have a large solution space. I need an algorithm that finds an equidistributed solution (that means equidistributed in the solution space) to that problem using a pseudo-random number generator (or fails).
Most standard algorithms for linear equation systems that I know from my linear algebra courses are not directly applicable to congruences as far as I can see...
My current, "safe" algorithm works as follows: Find all variable that appear in only one equation, and assign a random value. Now if in each row, only one variable is unassigned, assign the value according to the congruence. Otherwise fail.
Can anyone give me a clue how to solve this problem in general?
You can use gaussian elimination and similar algorithms just like you learned in your linear algebra courses, but all arithmetic is performed mod p (p is a prime). The one important difference is in the definition of "division": to compute a / b you instead compute a * (1/b) (in words, "a times b inverse"). Consider the following changes to the math operations normally used
addition: a+b becomes a+b mod p
subtraction: a-b becomes a-b mod p
multiplication: a*b becomes a*b mod p
division: a/b becomes: if p divides b, then "error: divide by zero", else a * (1/b) mod p
To compute the inverse of b mod p you can use the extended euclidean algorithm or alternatively compute b**(p-2) mod p.
Rather than trying to roll this yourself, look for an existing library or package. I think maybe Sage can do this, and certainly Mathematica, and Maple, and similar commercial math tools can.