Nonconvex Optimization via Joint Norm Relaxed SQP and Filled Function Method with Application to Minimax Two-Channel Linear Phase FIR QMF Bank Design - Advances in Heuristic Signal Processing and Applications - page 7

1.3 Joint Norm Relaxed Sequential Quadratic Programming
and Filled Function Method

A joint norm relaxed sequential quadratic programming and the filled function
method is proposed for finding the globally optimal solution of Problem ( ¯ ). The

details of the proposed method are discussed below.

1.3.1 Filled Function Method

Some terminologies related to filled functions are discussed below. Notably, a basin
of a function is defined as the subset of the domain of the optimization variables
such that any points in this subset will yield the same local minimum of the function
via conventional gradient based optimization methods. A hill of a function is defined
as the subset of the domain of the optimization variables such that any points in this
subset will yield the same local maximum of the function via conventional gradient
based optimization methods. A higher basin of a function is a basin of the function
with the objective functional value of the local minimum of the basin being higher
than that of the current basin of the function. A lower basin of a function is a basin
of the function with the objective functional value of the local minimum of the basin
being lower than that of the current basin of the function.
A filled function is a function satisfying the following properties: (a) the current
local minimum of the original objective function is the current local maximum of the
filled function; (b) the whole current basin of the original objective function is a part
of the current hill of the filled function; (c) the filled function has no stationary point
in any higher basins of the original objective function; and (d) there exists a local
minimum of the filled function which is in a lower basin of the original objective
function.
The working principles of the filled function method are as follows. Due to prop-
erty (a), by evaluating the filled function at a point slightly deviated from the current
local minimum of the original objective function, a lower functional value will be
obtained. Hence, the filled function could kick out from the current local minimum
of the original objective function. Due to properties (b)-(d), the current local min-
imum of the filled function is neither in the current basin nor in any higher basins
of the original objective function. Hence, the current local minimum of the filled
function is in a lower basin of the original objective function. As a result, by finding
the next local minimum of the original objective function via conventional gradient
based methods with the initial point being the current local minimum of the filled
function, a better local minimum of the original objective function can be obtained.
Following these procedures, if the original objective function contains a finite num-
ber of local minima, then the global minimum of the original objective function will
be eventually reached.
Based on the above working principles, the algorithm is summarized below.