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A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if there exist some numbers $\alpha$ and $\beta$ such that $a\le\alpha\le\beta\le b$ and the function is strictly monotone decreasing on $[a,\alpha]$, it is constant on $[\alpha,\beta]$, the function is strictly monotone increasing on $[\beta,b]$ and there is no stationary points outside $[\alpha,\beta]$. This property is very simple. In this paper, we show that a similar result holds for lower semicontinuous functions, which are pseudoconvex with respect to the lower Dini derivative. We prove that a function, defined on some interval, is pseudoconvex if and only if its domain can be split into three parts such that the function is strictly monotone decreasing in the first part, constant in the second one, strictly monotone increasing in the third part, and every stationary point is a global minimizer. Each one or two of these parts may be empty or degenerate into a single point. Some applications are derived., Comment: 9 pages, 1 figure

In this paper, we study some problems with continuously differentiable quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over the solution set and the normalized gradient is constant over it; (II) the gradient of the objective function is equal to zero over the solution set. As a consequence, we obtain characterizations of the solution set of a quasiconvex continuously differentiable program, provided that one of the solutions is known. We also derive Lagrange multiplier characterizations of the solutions set of an inequality constrained problem with continuously differentiable objective function and differentiable constraints, which are all quasiconvex on some convex set, not necessarily open. We compare our results with the previous ones. Several examples are provided., Comment: 15 pages

In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fr\'echet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A special class $\mathcal F_n$ of functions is defined and optimality conditions for isolated local minimum of order $n$ for a function $f\in\mathcal F_n$ are derived. The derivative of order $n$ does not appear in these characterizations. We prove necessary and sufficient criteria such that every stationary point of order $n$ is a global minimizer. We compare our results with some previous ones., Comment: 25 pages