##### Department of Mathematics,

University of California San Diego

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### Math 278C - Optimization and Data Science

## Uday Shanbhag

#### Pennsylvania State University

## Probability Maximization via Minkowski Functionals: Convex Representations and Tractable Resolution

##### Abstract:

In this talk, we consider the maximization of a probability $\mathbb{P}\{ \zeta \mid \zeta \in K(x)\}$ over a closed and convex set $\mathcal X$, a special case of the chance-constrained optimization problem. We define $K(x)$ as $K(x) \triangleq \{ {\zeta} \in к \mid c(x,\zeta) \geq 0 \}$ where $\zeta$ is uniformly distributed on a convex and compact set $к$ and $c(x,\zeta)$ is defined as either {$c(x,\zeta) \triangleq 1-|\zeta^Tx|m$, $m\geq 0$} (Setting A) or $c(x,\zeta) \triangleq Tx - \zeta$ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions, $\mathbb{P}\{ \zeta \mid \zeta \in K(x)\}$ can be expressed as the expectation of a suitably defined ${continuous}$ function $F({\bullet},\xi)$ with respect to an appropriately defined Gaussian density (or its variant), i.e. $\mathbb{E}_{\tilde p} [F(x,\xi)]$. Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of ${g(\mathbb{E} [F(x,\xi)])}$ over $\mathcal X$ where ${g}$ is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of ${g(\mathbb{E} [F(\bullet,\xi)]

To the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. Preliminary numerics on a portfolio selection problem (Setting A) and a vehicle routing problem (Setting B) suggest that the scheme competes well with naivemini-batch SA schemes as well as integer programming approximation methods. This is joint work with Ibrahim Bardakci, Afrooz Jalilzadeh, and Constantino Lagoa. Time permitting, a brief summary of ongoing work will be provided on ongoing research in hierarchical optimization and games under uncertainty.

Host: Jiawang Nie

### May 11, 2022

### 3:00 PM

https://ucsd.zoom.us/j/93696624146

Meeting ID: 936 9662 4146

Password: OPT2022SP

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