In this paper, a polymorphic uncertain nonlinear programming (PUNP)
model is constructed to formulate the problem of maximizing the
V-belt's fatigue life according to the practical engineering design
conditions. The model is converted into an equivalent interval
programming only involved with interval parameters for any given
degree of membership and confidence level. Then, a deterministic
equivalent formulation (DEF)
for the original model is obtained based on the concept of possibility degree for the order of two interval numbers. An algorithm, called sampling based algorithm, is developed to find a robust optimal design scheme for maximizing the fatigue life of the V-belt.
Case study is employed to demonstrate the validity and the
practicability of the constructed
model and the algorithm.
In this paper, we first present a concise representation of
multivariate polynomial, based on which we deduce the calculation
formulae of its derivatives using tensor. Then, we propose a
solution method to determine a global descent direction for the
minimization of general normal polynomial. At a local and non-global
maximizer or saddle point, we could use this method to get a global
descent direction of the objective function. By using the global
descent direction, we can transform an $n$-dimensional optimization
problem into a one-dimensional one. Based on some efficient
algorithms for one dimensional global optimization, we develop an
algorithm to compute the global minimizer of normal multivariate
polynomial. Numerical examples show that the proposed algorithm is
In this paper, a new spectral PRP conjugate gradient algorithm is
developed for solving nonconvex unconstrained optimization problems.
The search direction in this algorithm is proved to be a sufficient
descent direction of the objective function independent of line
search. To rule out possible unacceptably short step in the Armijo
line search, a modified Armijo line search strategy is presented.
The obtained step length is improved by employing the properties of
the approximate Wolfe conditions. Under some suitable assumptions,
the global convergence of the developed algorithm is established.
Numerical experiments demonstrate that this algorithm is promising.
When there is uncertainty in the lower level optimization problem of
a bilevel programming, it can be formulated by a robust optimization
method as a bilevel program with lower level second-order cone
programming problem (SOCBLP). In this paper, we show that the
Lagrange multiplier set mapping of the lower level problem of a
class of the SOCBLPs is upper semicontinuous under suitable
assumptions. Based on this fact, we detect the similarities and
relationships between the SOCBLP and its KKT reformulation. Then we
derive the specific expression of the critical cone at a feasible
point, and show that the second order sufficient conditions are
sufficient for the second order growth at an M-stationary point of
the SOCBLP under suitable conditions.
In this paper, a new definition of the filled function is given. Based on the new definition, a new class of filled functions is constructed, and the properties of the new filled functions are analysed and discussed. Moreover, according to the new class of filled functions, a criterion is given to decide whether the point we have obtained is an approximate global optimal solution. Finally, a global optimization algorithm based on the new class of filled functions is presented. The implementation of the algorithm on several test problems is reported with numerical results.
In this paper a filled function method is suggested for solving the
nonlinear complementarity problem. Firstly, the original problem is
converted into a corresponding unconstrained optimization problem by
using the Fischer-Burmeister function. Subsequently, a new filled
function with one parameter is proposed for solving unconstrained
optimization problems. Some properties of the filled function are
studied and discussed without Lipschitz continuity condition.
Finally, an algorithm based on the proposed filled function for
solving the nonlinear complementarity problem is presented. The
implementation of the algorithm on several test problems is reported
with numerical results.
We first propose an exact penalty method to solve strong-weak linear
bilevel programming problem (for short, SWLBP) for every fixed
cooperation degree from the follower. Then, we prove that the solution of
penalized problem is also that of the original problem under some
conditions. Furthermore, we give some properties of the optimal
value function (as a function of the follower's cooperation degree)
of SWLBP. Finally, we develop a method to acquire the critical
points of the optimal value function without enumerating all values
of the cooperation degree from the follower, and thus this function
is also achieved. Numerical results show that the proposed
methods are feasible.
The notions of
$\alpha$-well-posedness and generalized $\alpha$-well-posedness
for a system of constrained variational inequalities involving
set-valued mappings (for short, (SCVI)) are introduced in Hilbert spaces. Existence theorems of
solutions for (SCVI) are established by using penalty techniques.
Metric characterizations of $\alpha$-well-posedness and generalized
$\alpha$-well-posedness, in terms of the approximate solutions
sets, are presented. Finally, the equivalences between (generalized)
$\alpha$-well-posedness for (SCVI) and existence and uniqueness of
its solutions are also derived under quite mild assumptions.
In this paper, the behavior of the perturbation map is analyzed quantitatively by virtue of contingent derivatives and generalized contingent epiderivatives for the set-valued maps under strictly minimal efficiency. The purpose of this paper is to provide some well-known results concerning sensitivity analysis by applying a separation theorem for convex sets. When the results regress to multiobjective optimization, some related conclusions are obtained in a multiobjective programming problem.