The author considers an optimization problem which contains restrictions in the form of finitely many equalities and of inclusions involving an arbitrary convex body in a normed vector space, i.e. Q is a convex subset of a real vector space, H is a normed vector space, C is a convex body in H, \((\phi_ 0,\phi_ 1,\phi_ 2):Q\to {\mathbb{R}}\times {\mathbb{R}}^ m\times H\) and \(\bar q\) yields the minimum of \(\phi_ 0\) on the set \(\{q\in Q:\phi_ 1(q)=0,\phi_ 2(q)\in C\}.\) The second order conditions of the title are conditions in which different critical variations (\(y\in Q-\bar q\) is a critical variation if y satisfies \(\phi'_ 0(\bar q)y\leq 0\), \(\phi'_ 1(\bar q)y=0\), \(\phi'_ 2(\bar q)y\in C-\phi_ 2(\bar q))\) share a common Lagrange multiplier if they are ''pairwise critical'' (i.e. \(y_ 1,y_ 2\in Q-\bar q\), \(\phi''(\bar q)y_ 1y_ 2\leq 0\), \(\phi''_ 1(\bar q)y_ 1y_ 2=0\), \(\phi''_ 2(\bar q)y_ 1y_ 2\in C-\phi_ 2(\bar q))\). These conditions are more general than those in the author's previous paper [J. Differ. Equations. 28, 284-307 (1978; Zbl 0424.49011)] and in \textit{D. Bernstein's} paper [SIAM J. Control Optimization 22, 211-238 (1984)].