Well-Posedness of Optimization Problems

Stefan M. Stefanov ()
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Stefan M. Stefanov: South-West University Neofit Rilski

Chapter Chapter 9 in Separable Optimization, 2021, pp 169-180 from Springer

Abstract: Abstract Questions of existence of solutions and how they depend on problem’s parameters are usually important for many problems of mathematics, not only in optimization. The termWell-posedness of optimization problems well-posedness refers to the existence and uniqueness of a solution, and to its continuous behavior with respect to data perturbations, which is referred to as stability. In general, a problem is said to be stable ifStable problem $$ \varepsilon (\delta ) \rightarrow 0 \quad \mathrm{when} \quad \delta \rightarrow 0, $$ ε ( δ ) → 0 when δ → 0 , where $$\delta $$ δ is a given tolerance of the problem’s data, $$\varepsilon (\delta )$$ ε ( δ ) is the accuracy with which the solution can be determined, and $$\varepsilon (\delta )$$ ε ( δ ) is a continuous function of $$\delta $$ δ . Besides these conditions, accompanying robustness properties in the convergence of a sequence of approximate solutions are also required.

Date: 2021
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DOI: 10.1007/978-3-030-78401-0_9

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