The paper contains two main conclusions. In the first place there is a nonlinear Farkas-type theorem in which the objective function f(x): \(X\to {\mathbb{R}}\), \(f(0)=0\), is convex and continuous at 0 and the constraint g(x): \(X\to Y\) is S-sublinear, weakly \(S^*\)-lower semicontinuous with respect to a closed convex cone \(S\subset Y\). Both X and Y are real normed linear spaces. (A section at the beginning of the paper specifies notations and definitions.) This extends previous results of several authors in the sense that the convexity of f(x) replaces linearity or sublinearity assumptions. See results of \textit{B. D. Craven} [''Mathematical programming and control theory'' (1978; Zbl 0431.90039)] and \textit{B. M. Glover} [Z. Oper. Res., Ser. A 26, 125-141 (1982; Zbl 0494.90088)]. In the second part of the paper all the spaces are complete real normed linear. Again a Farkas-type result (Lemma 4.1) is obtained and from it several consequences in the form of alternative theorems, Lagrange conditions for optimality and existence of dual programs are derived. All the primal programs considered are of the form min\{f(x): -g(x)\(\in S\subset Y\), \(x\in C\subset X\}\), where f(x): \(X\to {\mathbb{R}}\) is convex and continuous at some point in the interior of the convex set C, the set S is a closed convex cone and g(x): \(X\to Y\) is weakly \(S^*\)-lower semicontinuous S-sublinear. Then, assuming the constraint qualification \(g(C)+S=Y\), Lemma 4.1 asserts that the two following statements are equivalent: (i) \(x\in C\), -g(x)\(\in S\Rightarrow f(x)\geq 0\) and (ii) there is \(\lambda \in S^*\) such that \(f(x)+\lambda g(x)\geq 0\) for all \(x\in C.\) \{Reviewer's remark: A nonlinear finite-dimensional predecessor of this lemma is due to A. Ghouila-Houri [\textit{C. Berge} and \textit{A. Ghouila- Houri}, ''Programming, games and transportation networks'' (French), Dunod, Paris (1962); English translation, see p. 67, Wiley, New York (1965)].\}