@@ -37,7 +37,7 @@ NLopt includes algorithms to attempt either *global* or *local* optimization of
**Local optimization** is the *much easier* problem of finding a feasible point **x** that is only a *local* minimum: *f*(**x**) is less than or equal to the value of *f* for all nearby feasible points (the intersection of the feasible region with at least some small neighborhood of **x**). In general, a nonlinear optimization problem may have *many* local minima, and which one is located by an algorithm typically depends upon the starting point that the user supplies to the algorithm. Local optimization algorithms, on the other hand, can often quickly locate a local minimum even in very high-dimensional problems (especially using gradient-based algorithms). (An algorithm that is guaranteed to find some *local* minimum from any feasible starting point is, somewhat confusingly, called *globally convergent*.)
In the special class of **convex optimization** problems, for which both the objective and inequality constraint functions are [convex](https://en.wikipedia.org/wiki/Convex_function)(and the equality constraints are [affine](https://en.wikipedia.org/wiki/Affine_transformation) or in any case have [convex](https://en.wikipedia.org/wiki/Convex_set)[level sets](https://en.wikipedia.org/wiki/Level_set)), there is only *one* local minimum value of *f* , so that a *local* optimization method finds a *global* optimum. There may, however, be more than one point **x** that yield the same minimum *f*(**x**), with the optimum points forming a convex subset of the (convex) feasible region. Typically, convex problems arise from functions of special analytical forms, such as linear programming problems, semidefinite programming, quadratic programming, and so on, and specialized techniques are available to solve these problems very efficiently. NLopt includes only general methods that do not assume convexity; if you have a provably convex problem, you may be better off with a different software package, such as the [CVX package](http://www.stanford.edu/~boyd/cvx/) from Stanford.
In the special class of **convex optimization** problems, for which both the objective and inequality constraint functions are [convex](https://en.wikipedia.org/wiki/Convex_function)(and the equality constraints are [affine](https://en.wikipedia.org/wiki/Affine_transformation) or in any case have [convex](https://en.wikipedia.org/wiki/Convex_set)[level sets](https://en.wikipedia.org/wiki/Level_set)), there is only *one* local minimum value of *f* , so that a *local* optimization method finds a *global* optimum. There may, however, be more than one point **x** that yield the same minimum *f*(**x**), with the optimum points forming a convex subset of the (convex) feasible region. Typically, convex problems arise from functions of special analytical forms, such as linear programming problems, semidefinite programming, quadratic programming, and so on, and specialized techniques are available to solve these problems very efficiently. NLopt includes only general methods that do not assume convexity; if you have a provably convex problem, you may be better off with a different software package, such as the [CVX package for Python](https://www.cvxpy.org/) or the [JuMP package for Julia](https://jump.dev/).
### Gradient-based versus derivative-free algorithms
