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Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new class of interest to the optimization community and practitioners, in which certain variables are required to be integers. In this paper, we describe five applications that lead to stochastic mixed-integer second-order cone programming problems. Additionally, we present solution algorithms for solving stochastic mixed-integer second-order cone programming using cuts and relaxations by combining existing algorithms for stochastic second-order cone programming with extensions of mixed-integer second-order cone programming. The applications, which are the focus of this paper, include facility location, portfolio optimization, uncapacitated inventory, battery swapping stations, and berth allocation planning. Considering the fact that mixed-integer programs are usually known to be NP-hard, bringing applications to the surface can detect tractable special cases and inspire for further algorithmic improvements in the future.

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature.