Rough Stability of Solutions to Nonconvex Optimization Problems

  • Hoang Xuan Phu
  • Hans Georg Bock
  • Sabine Pickenhain


The optimal solution set M(t) to some parametric optimization problem
$$\begin{array}{*{20}{c}} {minimize f(t,x)} & {subject to x \in D(t)} \\ \end{array}$$
is said to be roughly stable w.r.t. the roughness degree r > 0 at \( \bar t \in T\) if for all > 0 there is a neighborhood \( V\left( {\bar t} \right) \subset T\) of \( \bar t\) such that \( \left( {{ \cup _{t \in V\left( {\bar t} \right)}}M\left( t \right)} \right) < r + \in \) diam This paper states some sufficient conditions for this kind of generalized stability. One of the most important assumptions is that f is strictly roughly convexlike w.r.t. the roughness degree r. The result is applied to some optimal control problems, in particular, to a shipping problem.


Optimal Control Problem Transportation Cost Neighborhood Versus Shipping Problem Linear Normed Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Hoang Xuan Phu
    • 1
  • Hans Georg Bock
    • 2
  • Sabine Pickenhain
    • 3
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.IWR, University of HeidelbergHeidelbergGermany
  3. 3.Department of MathematicsTechnical University of CottbusCottbusGermany

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