In this talk, we study the stability of error bounds for the convex inequalities system under small perturbations and prove primal characterizations of the stability in terms of directional derivatives. For a single convex inequality, it is proved that the stability of error bounds is essentially equivalent to the non-zero minimum of the directional derivative at the reference points over the unit sphere. For the semi-infinite convex system, the stability of error bounds is essentially equivalent to verifying that the optimal values of several min-max problems defined by the directional derivatives of the component functions in the system

In this article, we portrays the global trading system as a network with countries as nodes and trade relations as edges. The general optimal equilibrium of the international trading network is obtained by constructing a nonlinear optimization model. Extreme shocks are simulated to illustrate the influence on the optimal equilibrium and suggestions are made to construct a more resilient trading network. Coping measures are proposed to alleviate pressures on the international trading network under the turbulent environment.

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics
affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.