The MOSEK command line tool Version 7.0 (Revision 141)

30 CHAPTER 4. PROBLEM FORMULATION AND SOLUTIONS
where
minimize
f(x) + c T x + c f
subject to l c ≤ g(x) + Ax ≤ u c ,
l x ≤ x ≤ u x ,
(4.17)
• m is the number of constraints.
• n is the number of decision variables.
• x ∈ R n is a vector of decision variables.
• c ∈ R n is the linear part objective function.
• A ∈ R m×n is the constraint matrix.
• l c ∈ R m is the lower limit on the activity for the constraints.
• u c ∈ R m is the upper limit on the activity for the constraints.
• l x ∈ R n is the lower limit on the activity for the variables.
• u x ∈ R n is the upper limit on the activity for the variables.
• f : R n → R is a nonlinear function.
• g : R n → R m is a nonlinear vector function.
This means that the ith constraint has the form
l c i ≤ g i (x) +
n∑
a ij x j ≤ u c i.
j=1
The linear term Ax is not included in g(x) since it can be handled much more efficiently as a separate
entity when optimizing.
The nonlinear functions f and g must be smooth in all x ∈ [l x ; u x ]. Moreover, f(x) must be a convex
function and g i (x) must satisfy
− ∞ < li c ⇒ g i (x) is concave,
u c i < ∞ ⇒ g i (x) is convex,
− ∞ < li c ≤ u c i < ∞ ⇒ g i (x) = 0.
4.5.1 Duality for general convex optimization
Similar to the linear case, MOSEK reports dual information in the general nonlinear case. Indeed in
this case the Lagrange function is defined by