In non-regular problems the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with conic constraints by Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, <CitationRef CitationID="CR17">2001</CitationRef>). They are based on the so-called 2-regularity condition of the...</citationref>

A function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${f : \Omega \to \mathbb{R}}$$</EquationSource> </InlineEquation> , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g − h with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${g, h : \Omega \to \mathbb{R}}$$</EquationSource> </InlineEquation> convex functions. While d.c. functions find various applications, especially in optimization, the...</equationsource></inlineequation></equationsource></inlineequation>

The problem of minimizing <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\tilde f=f+p}$$</EquationSource> </InlineEquation> over some convex subset of a Euclidean space is investigated, where f(x) = x <Superscript> T </Superscript> Ax + b <Superscript> T </Superscript> x is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> is strictly outer...</equationsource></inlineequation></superscript></superscript></equationsource></inlineequation>

A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ    0 if there is a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\nu \in]0, 1]$$</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f...</equationsource></inlineequation></equationsource></inlineequation>

A nonsmooth multiobjective optimization problem involving generalized (F, α, ρ, d)-type I function is considered. Karush–Kuhn–Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution. Duality results are...

The problem of minimizing $${\tilde f=f+p}$$ over some convex subset of a Euclidean space is investigated, where f(x) = x T Ax + b T x is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function $${\tilde f}$$ is strictly outer γ-convex...

A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ    0 if there is a $$\nu \in]0, 1]$$ such that $${\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f...