Quadratic Programming (QP) is a class of optimization problems that consist in maximizing a quadratic objective under linear constraints:

Min x_TMx    subject to    A_ix <= b_i

with   i=1,...,M  and x a vector of size N

QP is a subclass of SOCP (Second Order Cone Programs).

QP was first introduced in a portfolio selection context by Harry Markowitz (1952) in the Mean-Variance allocation method, where x is the vector of composition of the portfolio and M the variance-covariance matrix of the underlying components. 

The difficulty of solving the quadratic programming problem depends on the nature of the matrix M. In convex quadratic programs, such as the Mean-Variance method, the matrix M is positive semidefinite and the optimization problem is easy to solve with Quasi-Newton like algorithms. If M has negative eigenvalues, the quadratic program is nonconvex and then the objective function may have more than one local minimizer.