1. Optimization problem

Optimization problem set up with CPLEX

Aim: find a diet combination that satisfy the nutritional and environmental constraints, while similar to the current diet.

Notation

We make the following notation:

• \(x_1, x_2, ..., x_{k}\) are the target food intake (in grams, or other units) for \(k\) food groups.
• \(X_1, X_2, ..., X_{k}\) are the current food intake (in grams, or other units).

For the constraints,

• \(e_1, ..., e_k\): energy associated with each of the food groups
  • \(E\) is the total energy for all foods, with range between \(E_{lower}, E_{upper}\)
  • For example, with the data we have, this range is (9000, 10000).
• \(p_1, ..., p_k\): protein
• \(f_1, ..., f_k\): fat
• \(g_1, ..., g_k\): ghge

etc.

Optimization goal

Find a set of \(x_1, ..., x_k\) such that the values would

minimise the squared sum of differences between current diet and target diet:

\[(x_1 - X_1)^2 + (x_2 - X_2)^2 + ... + (x_k - X_k)^2\]

and satisfy the following constraints:

• \(x_1, ..., x_k >= 0\) (realistic diet intake can not be negative)
• \(x_1e_1 + x_2 e_2 + ... + x_k e_k >= E_{lower}\), total energy above the lower limit
• \(x_1e_1 + x_2 e_2 + ... + x_k e_k <= E_{upper}\), total energy below the upper limit
• \(x_1p_1 + x_2 p_2 + ... + x_k p_k >= P_{lower}\), total protein below the upper limit
• \(x_1p_1 + x_2 p_2 + ... + x_k e_k <= P_{upper}\), total protein below the upper limit

And so on.

Solve the optimization problem

This setting is a quadratic program (QP). It is an optimization problem with quadratic objective, and inequality constraints. We do not have equality constraints in this setting.

With R, there are various software to find a solution:

• CPLEX (chosen one)
• nloptr in nloptr package (non-linear optimization),
• constrOptim in stats package, which relies on optim function,
• solve.QP in quadprog package

among others.

CPLEX

Resources: official Rcplex vignette, page 2.

The CPLEX solver aims to solve the optimisation problem:

\[min \frac{1}{2} x' Q x + c'x\] subject to

\[Ax \leq b\]

\[lb \leq x \leq ub\]

The program in R looks like this

library(Rcplex)

# the values needs to be available
result_obj <- Rcplex(cvec = cvec,
                     Amat = Amat,
                     bvec = bvec,
                     Qmat = Qmat,
                     lb = lb,
                     ub = ub,
                     objsense = "min",
                     sense = sense)
# the result is saved in result_obj, to see the results
result_obj

1

cvec: vector, current diet values (multiplied by -2)

2

Amat: matrix, individual food contribution to total nutrition and environmental impact

3

bvec: vector, constraint values for nutrition and environmental impact for total diet

4

Qmat: matrix

5

lb, ub: lower and upper bound for target diet

6

objsense, sense: E, L, and R instead of ==, <= and >=