I am afraid the question is not very clear. exp(x) is a nonlinear function. By making it linear', do you mean you are looking for its linear approximation? This would depend on the domain of the problem. If is x is constrained to the neighbourhood of a known constant x0 (which may be zero, for example), you can use exp(x) \approx. exp(x0) + (x-x0)exp(x0). If x is an arbitrary real number, a linear approximation is not possible. Again, depending on other constraints etc, it MIGHT be possible to transform the problem into a linear one with a different set of variables (e.g., 2exp(x) is not linear in x, but is linear in y=exp(x). Any problem specified entirely in terms of exp(x) may be solvable as a linear problem in y and the solution can be inverted to find x).

Linear programming works actually for a special of non-linear problems. LP should be called piecewise linear convex programming.

If you are minimizing a non-linear but convex function (or maximizing a concave function) f(x) you can replace it with an arbitrarily accurate piecewise linear approximation and use LP.

To do this, a simple technique is to represent the function as a convex combination (weighted average) of coordinate pairs (x_i, f_i) where f_i = f(x_i), i= 0, ...,N.

Then you choose weights w_i for each coordinate pair and instead of f(x) in the objective function you write sum(i= 0,...,N, w_i*f_i).

Then you add constraints

x=sum(i=0,...,N, w_i*x_i);

sum(i=0,...,N, w_i)=1;

w_i>=0;

Same works also in multiple dimensions, i.e. functions with more arguments.

To use this technique in constraints, the constraint must be a convex constraint, i.e. f(x) must be convex and appear on the left hand side of a less-than-or-equal-to constraint

f(x) + ...

In addition to linear approximation, you may be able to achieve it through discretization of x . In many practical applications, discretization of x yields a good solution without much of a compromise on the solution quality. However, we need to understand the detailed formulation to see if discretization works without variable explosion.

You can write the taylor series expansion and then only take the first 2 terms to make it linear.

~SK

That depends on the form of your objective function, if exp(x) is a factor multiplied by some other explicit factor, you may transform to the logarithm to make it linear; if objective function is too complicated, i think the best solution is to apply taylor expansion as Srikanth suggests.

Hi,

Many thanks for your answers. But I have to emphasize that 'x' has lower and upper bounds (i.e., a

To augment my previous answer, applying piecewise linear approximations for non-convex problems, it is straight-forward by MILP encoding. 

Define binary (0/1) variables y_i for each line segment between points i-1 and i, and make their sum equal to 1 by adding constraint sum(i=1,...,N, y_i)=1. Then add constraints to allow only the weights corresponding to end-points of that line segment to be non-zero: w_i

In some situations introducing new variable z=exp(x) may be helpful.  But I guess your case is not susceptible to such a simple trick.