Solver interfaces are implemented as "plug-ins". A new plugin [2] allows to run solvers on NEOS [3]. This gives us access to a number of powerful solvers, without the need to install things locally.

#### Linear models

To illustrate how we can express a simple transportation model in ROI, let's try a very simple transportation model [4]: \[\bbox[lightcyan,10px,border:3px solid darkblue]{\begin{align}\min\>&\sum_{i,j} c_{i,j} x_{i,j}\\&\sum_j x_{i,j} \le a_i\>\>\forall i\\&\sum_i x_{i,j} \ge b_j\>\>\forall j\\&x_{i,j}\ge 0\end{align}}\]

The GAMS representation is:

Set i 'canning plants' / seattle, san-diego / j 'markets' / new-york, chicago, topeka /; Parameter a(i) 'capacity of plant i in cases' / seattle 350 san-diego 600 / b(j) 'demand at market j in cases' / new-york 325 chicago 300 topeka 275 /; Table d(i,j) 'distance in thousands of miles' new-york chicago topeka seattle 2.5 1.7 1.8 san-diego 2.5 1.8 1.4; Scalar f 'freight in dollars per case per thousand miles' / 90 /; Parameter c(i,j) 'transport cost in thousands of dollars per case'; c(i,j) = f * d(i,j) / 1000; Variable x(i,j) 'shipment quantities in cases' z 'total transportation costs in thousands of dollars'; Positive Variable x; Equation cost 'define objective function' supply(i) 'observe supply limit at plant i' demand(j) 'satisfy demand at market j'; cost.. z =e= sum((i,j), c(i,j)*x(i,j)); supply(i).. sum(j, x(i,j)) =l= a(i); demand(j).. sum(i, x(i,j)) =g= b(j); Model transport / all /; solve transport using lp minimizing z; display x.l, x.m;

Note that all vectors and matrices have strings as indices. The ordering does not matter (like in a relational database table). Now let's do the same for ROI.

# # data # n <- 2 # number of plants (sources) m <- 3 # number of markets (distinations) si <- c('seattle', 'san-diego') sj <- c('new-york', 'chicago', 'topeka') a <- c(350, 600) # capacity b <- c(325, 300, 275) # demand # distances d <- matrix(c(2.5, 1.7, 1.8, 2.5, 1.8, 1.4), nrow=n, ncol=m, byrow=TRUE) f <- 90 # unit freight cost c <- d*f/1000 # unit transportation cost # flatten c row wise cflat <- as.vector(t(c)) # create constraint matrix .... somewhat torturous code nvar <- n * m; A <- numeric(0) varnames <- character(0); for (i in 1:n) { row <- rep(0,nvar) for (j in 1:m) { row[(i-1)*m+j] <- 1 varnames <- c(varnames,paste(si[i],sj[j],sep=":")) } A<-rbind(A,row) } for (j in 1:m) { row <- rep(0,nvar) for (i in 1:n) row[(i-1)*m+j] <- 1 A<-rbind(A,row) } rhs <- c(a,b) dir <- c(rep('<=',n),rep('>=',m)) LP <- OP(L_objective(cflat,names=varnames), L_constraint(A,dir,rhs), max=FALSE) (sol <- ROI_solve(LP, solver = "glpk"))

I just used vectors with integer indices: i.e. the position determines the meaning of a number (in other words: the ordering is important). We could have used vectors/matrices with row and column names, although I am not sure if this would make things much more readable.

The results look like:

> (sol <- ROI_solve(LP, solver = "glpk")) Optimal solution found. The objective value is: 1.536750e+02 > solution(sol) seattle:new-york seattle:chicago seattle:topeka san-diego:new-york san-diego:chicago san-diego:topeka 50 300 0 275 0 275

When reading the GAMS code, you can follow this is about a transportation model. The R code is much more obscure, it is close to write-only code. I don't understand how you can implement large and/or complex models in this way.

The R code builds up a "large" matrix \(A\). For this tiny example it is small, but in general it is very large: in practice even way too large to handle even medium sized models. A model with 10,000 equations and 10,000 variables (not very large by today's standards) leads to a 10,000 x 10,000 matrix with 100 million entries! This type of matrix based interface essentially brings us back to the days of matrix generators [7] (although they paid attention to sparsity). This technology is largely obsolete.

#### NEOS

To use a NEOS solver we just need to change the last line:

> (sol <- ROI_solve(LP, solver = "neos", method = "mosek")) Optimal solution found. The objective value is: 1.536750e+02

For some solvers you need to provide an e-mail address:

In this case, add email="...".

#### Behind the scenes

Interestingly, under the hood, a GAMS model is generated and sent to NEOS. It looks like:

> model_call <- ROI_solve(LP, solver = "neos", method = "mosek", + dry_run = TRUE) > cat(as.list(model_call)$xmlstring) <?xml version="1.0" encoding="UTF-8"?> <document> <category>milp</category> <solver>MOSEK</solver> <inputMethod>GAMS</inputMethod> <model><![CDATA[Option IntVarUp = 0; Set i / R1*R5 / ; Set ileq(i) / R1, R2 / ; Set igeq(i) / R3, R4, R5 / ; Set j / C1*C6 / ; Parameter objL(j) /C1 0.225 C2 0.153 C3 0.162 C4 0.225 C5 0.162 C6 0.126/ ; Parameter rhs(i) /R1 350 R2 600 R3 325 R4 300 R5 275/ ; Parameter A /R1.C1 1 R3.C1 1 R1.C2 1 R4.C2 1 R1.C3 1 R5.C3 1 R2.C4 1 R3.C4 1 R2.C5 1 R4.C5 1 R2.C6 1 R5.C6 1/; Variables obj; Positive Variables x(j); Equations ObjSum LinLeq(ileq) LinGeq(igeq); ObjSum .. obj =e= sum(j, x(j) * objL(j)) ; LinLeq(ileq) .. sum(j, A(ileq, j) * x(j)) =l= rhs(ileq) ; LinGeq(igeq) .. sum(j, A(igeq, j) * x(j)) =g= rhs(igeq) ; Model LinearProblem /all/ ; Solve LinearProblem using LP minimizing obj ; option decimals = 8; display '---BEGIN.SOLUTION---', x.l, '---END.SOLUTION---'; file results /results.txt/; results.nw = 0; results.nd = 15; results.nr = 2; results.nz = 0; put results; put 'solution:'/; loop(j, put, x.l(j)/); put 'objval:'/; put LinearProblem.objval/; put 'solver_status:'/; put LinearProblem.solvestat/; put 'model_status:'/; put LinearProblem.modelstat/;]]></model> <options><![CDATA[]]></options> <gdx><![CDATA[]]></gdx> <wantgdx><![CDATA[]]></wantgdx> <wantlog><![CDATA[]]></wantlog> <comments><![CDATA[]]></comments> </document> >

All structure is lost: we are just modeling:\[\begin{align} \min \>& c^Tx \\ & A_1 x \le b_1 \\& A_2 x \ge b_2 \end{align}\]

#### Quadratic models

This plugin also allows quadratic models (but not general nonlinear models). In [2] a trivial non-convex QP model is presented: \[\begin{align}\max\>&F(P_F - 150) + C(P_C - 100) \\ & 2 F + C \le 500\\ & 2F + 3C \le 800 \\ & F \le 490 - P_F \\ & C \le 640 - 2P_C\\ & F, C, P_F, P_C\ge 0\end{align}\] ROI does not understand mathematics, so we have to pass this on in terms of matrices:

Q <- simple_triplet_matrix(i = 1:4, j = c(2, 1, 4, 3), rep(1, 4)) as.matrix(Q) var_names <- c("full", "price_full", "compact", "price_compact") o <- OP( Q_objective(Q = Q, L = c(-150, 0, -100, 0), names = var_names), L_constraint(rbind(c(2, 0, 1, 0), c(2, 0, 3, 0), c(1, 1, 0, 0), c(0, 0, 1, 2)), dir = leq(4), rhs = c(500, 800, 490, 640)), maximum = TRUE) (sol <- ROI_solve(o, solver = "neos", method = "BARON"))

Unfortunately, this is not really close to the mathematical model and makes the problem setup difficult to read and understand.

In [2] a table is presented with some of the supported solvers:

#### Linear models: OMPR

For linear models, we can use the OMPR package [5,6] on top of ROI. This will give us more readable models. Here we implement the above transportation model. The model can look like:

library(ompr) library(ompr.roi) library(dplyr) # # data # n <- 2 # number of plants (sources) m <- 3 # number of markets (distinations) si <- c('seattle', 'san-diego') sj <- c('new-york', 'chicago', 'topeka') a <- c(350, 600) # capacity b <- c(325, 300, 275) # demand # distances d <- matrix(c(2.5, 1.7, 1.8, 2.5, 1.8, 1.4), nrow=n, ncol=m, byrow=TRUE) f <- 90 # unit freight cost c <- d*f/1000 # unit transportation cost # # model + solve # (sol <- MIPModel() %>% add_variable(x[i,j],i=1:n,j=1:m,type = "continuous", lb = 0) %>% set_objective(sum_expr(c[i,j]*x[i,j],i=1:n,j=1:m), "min") %>% add_constraint(sum_expr(x[i,j],j=1:m)<=a[i], i=1:n) %>% add_constraint(sum_expr(x[i,j],i=1:n)>=b[j], j=1:m) %>% solve_model(with_ROI(solver = "neos", method="mosek"))) # # solution # sol %>% get_solution(x[i, j]) %>% filter(value > 0)

This code at least has some resemblance to the mathematical model. The solution looks like:

Status: optimal Objective value: 153.675 variable i j value 1 x 1 1 50 2 x 2 1 275 3 x 1 2 300 4 x 2 3 275

We can clean up the solution a bit, so we get a more meaningful solution report:

> sol %>% + get_solution(x[i, j]) %>% + filter(value > 0) %>% + mutate(Plant = si[i],Market = sj[j]) %>% + select(Plant,Market,Shipment=value) Plant Market Shipment 1 seattle new-york 50 2 san-diego new-york 275 3 seattle chicago 300 4 san-diego topeka 275

#### But..

OMPR is much better than matrix oriented tools to express models. That does not mean OMPR is suited for all LP/MIP models. For one thing it can be slow, there is no good support for large,sparse structures and there are some bugs. Here is a recent example of a bug I encountered.

In many cases we may need to restrict the domain of variables. E.g. instead of all

**x[i,j]**, only use a subset. An example is a network of nodes and arcs. The arcs can be represented by**x[i,j]**indicating the flow from node**i**to node**j**. Of course, if we have a sparse network only a few of all**(i,j)**combinations are allowed.
OMPR allows conditions on the domain of variables. E.g. for a one-dimensional variable

**x[i]**, we can do:
>
n <- 3

>
ok <- c(1,0,1)

>
ok

[1]
1 0 1

>
(sol <- MIPModel() %>%

+ add_variable(x[i],
i=1:n, ok[i]==1)

+
)

Mixed
integer linear optimization problem

Variables:

Continuous:
2

Integer:
0

Binary:
0

No
objective function.

Constraints:
0

This looks great. We see

**x[2]**is not included in the variables. Now let's do the same thing for a two dimensional variable

**x[i,j]**:

> n <- 3

> ok <- matrix(c(0,1,1, 0,0,1, 0,0,0),nrow=3,ncol=3,byrow=T)

> ok

` [,1] [,2] [,3]`

`[1,] 0 1 1`

`[2,] 0 0 1`

`[3,] 0 0 0`

> (sol <- MIPModel() %>%

+ add_variable(x[i,j], i=1:n, j=1:n, ok[i,j]==1)

+ )

`Error in validate_quantifier_candidates(candidates, zero_vars_msg) : `

` only_integer_candidates are not all TRUE`

Something goes wrong here. The error message is not very informative: I have no clue what it means. I also have no idea how to fix this.

#### References

- Many Solvers, One Interface, ROI, the R Optimization Infrastructure Package, https://www.r-project.org/conferences/useR-2010/slides/Theussl+Hornik+Meyer.pdf
- Ronald Hochreiter and Florian Schwendinger, ROI Plug-in NEOS, https://cran.r-project.org/web/packages/ROI.plugin.neos/vignettes/ROI.plugin.neos_Introduction.pdf
- NEOS Server: State-of-the-Art Solvers for Numerical Optimization, https://neos-server.org/neos/
- Matlab vs GAMS: linear programming, http://yetanothermathprogrammingconsultant.blogspot.com/2016/09/matlab-vs-gams-linear-programming.html
- Mixed Integer Linear Programming in R, https://dirkschumacher.github.io/ompr/
- MIP model in R using the OMPR package, http://yetanothermathprogrammingconsultant.blogspot.com/2017/04/mip-model-in-r-using-ompr-package.html
- Robert Fourer, Modeling Languages versus Matrix Generators for Linear Programming, ACM Transactions on Mathematical Software, vol. 9, pp. 143–183, 1983.