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We discuss an application of the well-known Multiplicative Weights Update (MWU) algorithm to non-convex and mixed-integer nonlinear programming. We present applications to: (a) the distance geometry problem, which arises in the positioning of mobile sensors and in protein conformation; (b) a hydro unit commitment problem arising in the energy industry, and (c) a class of Markowitz' portfolio selection problems. The interest of the MWU with respect to one of its closest competitors (classic Multi-Start) is that it provides a relative approximation guarantee on a certain quality measure of the solution. Algorithm 1 Multi-Start 1: while termination condition is not met do 2: sample a starting point x 3: perform a local descent from x , yieldingx 4:ifx improves the best optimum x * so far, update x * withx 5: end while We must determine the best distribution to sample from, what kind of local descent algorithm to employ,

In this paper, we review convex mixed-integer quadratic programming approaches to deal with single-objective single-period mean-variance portfolio selection problems under real-world financial constraints. In the first part, after describing the original Markowitz's mean-variance model, we analyze its theoretical and empirical limitations, and summarize the possible improvements by considering robust and probabilistic models, and additional constraints. Moreover, we report some recent theoretical convexity results for the probabilistic portfolio selection problem. In the second part, we overview the exact algorithms proposed to solve the single-objective single-period portfolio selection problem with quadratic risk measure.

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