rpsg_riskconstrprog: Solves risk minimization problems with constraint on a risk...

In PSGExpress: Portfolio Safeguard: Optimization, Statistics and Risk Management

Description

Finds a minimum for the problem

min_x [risk1(x) + d\cdot x]

subject to

risk2 ≤q r

A \cdot x= b

Aeq \cdot x= beq

lb ≤q x ≤q ub

where
A, Aeq are matrices;
d, x are vectors;
b, beq, lb, ub are vectors or scalars;
risk1(x), risk2(x) functions are linear combinations of PSG risk functions, PSG deterministic functions, or PSG utility functions. List of PSG functions for riskprog

Usage

rpsg_riskconstrprog(model, stroptions = NULL)

Arguments

model

list with data for the optimization problem. Some components are optional. Names of list members: model$risk1 character with description of the PSG function (objective includes only one PSG function) or character vector with coefficients and PSG functions in one of the following forms: coefficient 1, function 1, ... , coefficient K, function K; "na" denotes the absent parameter. List of PSG functions for riskconstrprog model$risk2 character with description of the PSG function (constraint includes only one PSG function) or character vector with coefficients and PSG functions in one of the following forms: coefficient 1, function 1, ... , coefficient K, function K; "na" denotes the absent parameter. List of PSG functions for riskconstrprog model$w1 parameter (numeric) of the PSG function (for one PSG function in objective) or vector of parameters (for linear combination of PSG functions; order of parameters are the same as the order of functions in model$risk1, "na" denotes the absent parameter); model$w2 parameter (numeric) of the PSG function (for one PSG function in constraint) or vector of parameters (for linear combination of PSG functions; order of parameters are the same as the order of functions in model$risk2, "na" denotes the absent parameter); model$H1 matrix for one PSG function in objective or vector of matrices for linear combination of PSG functions model$risk1 (order of matrix are the same as the order of functions in model$risk1); model$H2 matrix for one PSG function in constraint or vector of matrices for linear combination of PSG functions model$risk2 (order of matrix are the same as the order of functions in model$risk2); model$c1 vector of benchmark for one PSG function in objective or vector of vectors of benchmarks for linear combination of PSG functions (order of vectors are the same as the order of functions in model$risk1); model$c2 vector of benchmark for one PSG function in constraint or vector of vectors of benchmarks for linear combination of PSG functions (order of vectors are the same as the order of functions in model$risk2); model$p1 vector of probabilities for one PSG function in objective or vector of vectors of probabilities for linear combination of PSG functions (order of vectors are the same as the order of functions in model$risk1); model$p2 vector of probabilities for one PSG function in constraint or vector of vectors of probabilities for linear combination of PSG functions (order of vectors are the same as the order of functions in model$risk2); model$d vector for linear component of objective; model$rineq value of upper bound for risk constraint; model$Aineq matrix for linear inequality constraint; model$bineq vector or scalars for linear inequality constraint; model$Aeq matrix for linear equality constraint; model$beq vector for linear equality constraint; model$lb vector of lower bounds for x; model$ub vector of upper bounds for x; model$x0 initial point for x.

stroptions

list with additional optimization options: stroptions$solver character with name of optimization solver: VAN (default), CAR, BULDOZER, TANK; stroptions$precision number of digits that solver tries to obtain in objective and constraints (default = 7); stroptions$time.limit time in seconds restricting the duration of the optimization process; stroptions$linearization1 number 0 or 1, controls internal representation of risk function in objective (model$risk1), which can speed up the optimization process (used with CAR and TANK solvers); stroptions$linearization2 number 0 or 1, controls internal representation of risk function in constraint (model$risk2), which can speed up the optimization process (used with CAR and TANK solvers); stroptions$stages number of stages of the optimization process. This parameter should be specified for VaR, Probability, and Cardinality groups of functions, default = 9; stroptions$types number that specifies the variable types of a problem. If Types is defined as column-vector, it should include as many components as number of variables Problem includes. The components of column-vector can possess the values 0 - for variables of type real, or 1- for variables of type boolean, or 2 - for variables of type integer. If Types is defined as one number (0, or 1, or 2) than all variable types are real, or boolean, or integer respectively; stroptions$mip1 number that specifies the linearization of functions in objective (model$risk1) using MIP capabilities of Gurobi. Active only in VANGRB, CARGRB or HELI solvers (which are based on Gurobi optimization solvers and are available only if Gurobi is installed). stroptions$mip2 number that specifies the linearization of functions in constraint (model$risk2) using MIP capabilities of Gurobi. Active only in VANGRB, CARGRB or HELI solvers (which are based on Gurobi optimization solvers and are available only if Gurobi is installed). stroptions$save.to.text character with path to the folder for storing problem in General (Text) Format of PSG.

Value

list results with solution results:

results$status

status of solved problem;

results$objective

optimal value of objective function;

results$gap

difference between objective value in obtained point and Lower estimate of optimal value;

results$optimal.point

optimal point;

results$risk.constraint.value

optimal values of left hand sides of risk constraint;

results$risk.constraint.residual

residual of risk constraint;

results$ineq.constraint.value

optimal values of left hand sides of linear inequality constraint;

results$ineq.constraint.residual

residual of linear inequality constraint;

results$eq.constraint.value

optimal values of left hand sides of linear equality constraint;

results$eq.constraint.residual

residual of linear equality constraint;

results$function.value

optimal values of PSG functions defined in problem;

results$loading.time

data loading time;

results$preprocessing.time

preprocessing time;

results$solving.time

solving time.

Note

• Exponential Utility, Logarithmic Utility, Power Utility, Logarithms Sum, and Logarithms Exponents Sum functions are included in the objective with the negative coefficient -1 by default. However, you should not set any coefficient for these functions, riskprog sets it automatically.

• linear, pr_dev, pr_dev_g, pr_pen, and pr_pen_g PSG functions should NOT be combined with the vector d.

• Relative Entropy function (entropyr) can have up to 100,000,000 decision variables if stroptions$linearization=1 is specified. This option may dramatically speedup calculations. In this case,a BULDOZER solver is recommended (options$solver <- "buldozer").

Author(s)

Stan Uryasev [aut, cre, cph], Grigoriy Zrazhevsky [aut], Viktor Kuzmenko [aut], Alex Zrazhevsky [aut]

Maintainer: Stan Uryasev <[e-mail]>

References

American Optimal Decisions
Portfolio Safeguard Help

See Also

rpsg_riskprog rpsg_riskratioprog

Examples

#Problem of CVaR minimization with constraint on the mean profit:
#Find x = (x1,x2,x3,x4) minimizing
#risk(x) = CVaR(0.95,x)
#subject to
#Average Gain(x)>4.5
#x1+x2+x3+x4 = 1
#x1>=0, x2>=0, x3>=0, x4>=0

input.model <- list()
input.model$risk1 <- "cvar_risk"
input.model$w1 <- 0.95
input.model$H1<-matrix(c(1,4,8,3, 7,5,4,6, 2,8,1,0,0,3,4,9),nrow=4, byrow=TRUE)
input.model$c1 <- c(0.2, 0.11, 0.6, 0.1)
input.model$risk2 <- "avg"
input.model$H2<-matrix(c(1,4,8,3, 7,5,4,6, 2,8,1,0,0,3,4,9),nrow=4, byrow=TRUE)
input.model$c2 <- c(0.2, 0.11, 0.6, 0.1)
input.model$rineq <- -4.5
input.model$Aeq <- matrix(c(1, 1, 1, 1),nrow=1)
input.model$beq <- 1
input.model$lb<-c(0, 0, 0, 0)

options<-list()
options$solver <- "van"
options$precision <- 7
options$stages <- 30

results <- rpsg_riskconstrprog(input.model,options)

print(results)

PSGExpress documentation built on July 26, 2019, 5:02 p.m.