Second-time planning

Two-time planning issues
is a typical optimization problem, including convex two-time planning and non-convex two-time programming, in which the objective function is the two-time function of the variable, and the constraint condition is the linear inequality of the variable.

Assuming that the number of variables is D D, the number of constraints is M m, then the standard two-time planning problem is shaped like: Minxs.t.12xtqx+ctxax⩽b \begin{matrix} \min_{x} &\frac{1}{2}x^tqx+c^tx\\ S.t.&ax \leqslant b \end{matrix}
where x x is d d-dimensional vector, q∈rdxd q\in \mathbb{r^{d\times D}} is a real symmetric matrix, a∈rmxd a\in \mathbb{r^{m \times D}} is a real matrix, b∈rm b\in \mathbb{r^m } and C∈rd c\in \mathbb{r^d} are real vectors, and each line of ax⩽b Ax \leqslant b corresponds to a constraint.

If q Q is a semi-positive definite matrix, the above objective function is a convex function, and the corresponding two-time plan is convex two-time programming problem; If the constraint defines a feasible field that is not empty and the target function has a lower bound in this feasible field, the problem has a global minimum. If q q is a positive definite matrix, then the problem has a unique global minimum value. If q Q is a non-positive definite matrix, the objective function is a NP-hard problem with multiple stationary points and local minima.

Common methods for solving two-time programming problems are: Ellipsoid method, augmented Lagrange method, gradient projection method.
such as If q q is a positive definite matrix, then the corresponding two-time programming problem can be solved by the ellipsoid method in polynomial time.

convex function:

The function f F, defined on the interval [a, b] [b], if it is to any of the two-point x1,x2 x_1,x_2 in the interval: F (x1+x22