The Kuhn-Tucker and envelope theorems can be used to characterize the solution to a wide range of constrained optimization problems: static or dynamic, under perfect foresight or featuring randomness and uncertainty. In addition, these same two results provide foundations for the work on the maximum principle and dynamic programming that we will do later on. For both of these reasons, the Kuhn-...

The Thomas-Reiche-Kuhn sum rule is a fundamental consequence of the position-momentum commutation relation for an atomic electron and it provides an important constraint on the transition matrix elements for an atom. Analogously, the commutation relations for the electromagnetic field operators in a magnetodielectric medium constrain the properties of the dispersion relations for the medium thr...

Kuhn and Feyerabend have little to say about the thought of Michael Polanyi, and the secondary literature on Polanyi’s relation to them is meagre. I argue that Polanyi’s view, in Personal knowledge and in other writings, of conceptual frameworks ‘segregated’ by a ‘logical gap’ as giving rise to controversies in science foreshadowed Kuhn and Feyerabend’s theme of incommensurability. The similari...

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...

In one of his fi rst papers, Thomas Kuhn ( 1959 ) addressed the “essential tension” implicit in scientifi c research, i.e., the contrast between convergent and divergent thinking. He considered both to be central to the advance of science. Convergent thinking is what scientists do in their daily “normal research projects,” where the “scientist is not an innovator but a solver of puzzles, and th...

When multiple followers are involved in a bilevel decision problem, the leader’s decision will be affected, not only by the reactions of these followers, but also by the relationships among these followers. One of the popular situations within this framework is where these followers are uncooperatively making decisions while having cross reference of decision information. This situation is call...