Approximating Multiple Robust Optimization Solutions in One Pass via Proximal Point Methods

## Abstract

Abstract Robust optimization provides a principled and unified framework to model many problems in modern operations research and computer science applications, such as risk measures minimization and adversarially robust machine learning. To use a robust solution (e.g., to implement an investment portfolio or perform robust machine learning inference), the user has to a priori decide the trade-off between efficiency (nominal performance) and robustness (worst-case performance) of the solution by choosing the uncertainty level hyperparameters. In many applications, this amounts to solving the problem many times and comparing them, each from a different hyperparameter setting. This makes robust optimization practically cumbersome or even intractable. We present a novel procedure based on the proximal point method (PPM) to efficiently approximate many Pareto efficient robust solutions at once. This effectively reduces the total compute requirement from N × T 𝑁 𝑇 N\times T italic_N × italic_T to 2 × T 2 𝑇 2\times T 2 × italic_T , where N 𝑁 N italic_N is the number of robust solutions to be generated, and T 𝑇 T italic_T is the time to obtain one robust solution. We prove this procedure can produce exact Pareto efficient robust solutions for a class of robust linear optimization problems. For more general problems, we prove that with high probability, our procedure gives a good approximation of the efficiency-robustness trade-off in random robust linear optimization instances. We conduct numerical experiments to demonstrate.

## 1 Introduction

One of the main obstacles in deploying robust optimization models for real-world decision-making under uncertainty is determining an appropriate trade-off between nominal performance and worst-case performance before the uncertainty is realized. For example, in portfolio optimization, setting the level of risk aversion is challenging. Similarly, in adversarial machine learning, choosing the adversarial perturbation set parameters to balance average accuracy versus accuracy under adversarial attacks is a critical decision. Decision makers often need to obtain and test multiple robust solutions, each corresponding to a different solution on the efficiency-robustness Pareto frontier, before deciding which one to deploy.

To obtain multiple robust solutions, decision makers adjust hyperparameters such as the shape and size (radius) of the uncertainty set (Soyster, ; Ben-Tal and Nemirovski, ; Bertsimas and Sim, ) or by employing globalized robust optimization approaches that use penalty functions and coefficient hyperparameters (Ben-Tal et al., ; ). Consequently, generating each robust solution demands solving a different instance of the robust optimization problem, which is generally more computationally expensive than solving their deterministic counterparts. For instance, adversarial training is more expensive than traditional training because it involves solving an inner maximization problem to find adversarial attacks. In theory, the robust counterpart of linear programs (LPs) with ellipsoidal uncertainty sets becomes second-order cone programs (SOCPs), and the robust counterpart of uncertain SOCPs becomes semidefinite programs (SDPs). As a result, finding the right robust solution can be prohibitively costly in practice, let alone finding many of them to compare.

To address this challenge, we propose a new way of thinking about this problem, along with a novel proximal point method (PPM)-based algorithm to approximate efficiency-robustness Pareto efficient robust solutions, reducing the computational cost from $N\times T$ to $2\times T$, where $N$ is the number of different robust solutions on the Pareto frontier and $T$ is the cost of solving a single instance of a robust optimization problem. Specifically, the procedure first solves for the “most-robust” solution, then uses it as the starting point to perform proximal point updates towards the “least-robust” (deterministic counterpart) solution. We discover, intriguingly, the proximal point trajectory approximates the set of Pareto efficient robust solutions that we want to obtain in the first place.

We prove that for robust LPs with uncertain objective functions under the simplex decision domain and ellipsoidal uncertainty sets, the proximal point trajectory are *exactly* Pareto efficient robust solutions. For robust LPs with a random polyhedron domain, we prove that with high probability, the performances of the Pareto efficient robust solutions are bounded by the performances of two proximal point trajectories. To validate, we conduct numerical experiments at the end on portfolio optimization and adversarially robust deep learning.

## 2 Problem Setting and Main Idea

### 2.1 Notations

We denote the dual norm of $\|\cdot\|$ as $\|\cdot\|_{*}$, defined as $\|z\|_{*}=\sup\{z^{\top}x:\|x\|\leq 1\}$. The $p$-norm, $\|\cdot\|_{p}$, is given by $\|x\|_{p}=\left(\sum^{n}_{i=1}|x_{i}|^{p}\right)^{1/p}$. The infinity norm, $\|\cdot\|_{\infty}$, is defined as $\|x\|_{\infty}=\max_{i\in[n]}|x_{i}|$. The ball of radius $r$ around $x$ with respect to the norm $\|\cdot\|$ is represented by $\mathbb{B}_{\|\cdot\|}(x,r)=\{z\in\mathbb{R}^{n}:\|z-x\|\leq r\}$, and the ball of radius $r$ around the origin is denoted as $\mathbb{B}_{\|\cdot\|}(r)=\{z\in\mathbb{R}^{n}:\|z\|\leq r\}$. The projection operator onto the set $\mathcal{X}$ is denoted as $\Pi_{\mathcal{X}}(x)=\arg\min_{x^{\prime}\in\mathcal{X}}\|x^{\prime}-x\|^{2}_{2}$. Finally, $e$ represents a vector of ones.

### 2.2 Problem Setting

We start with a constrained optimization problem with parameters $a$ in the objective function:

$$ $\quad\min_{x\in\mathcal{X}}f(x,a).$ $$

The decision variable $x\in\mathbb{R}^{n}$ is subject to a compact and convex domain $\mathcal{X}$, $a\in\mathbb{R}^{n}$ is an uncertain vector, which is only known to belong to an uncertainty set $\mathcal{U}$. The robust counterpart is

$$ $\text{(RC)}\quad\min_{x\in\mathcal{X}}\max_{a\in\mathcal{U}}f(x,a),$ (1) $$

where the uncertainty set $\mathcal{U}$ takes the following form:

$$ $\mathcal{U}=\left\{a_{0}+\xi:\ \xi\in\Xi\subset\mathbb{R}^{n}\right\}.$ $$

Here $a_{0}$ is some fixed nominal vector, $\xi$ can be interpreted as the perturbation, and $\Xi$ is a nonempty compact convex set. In the case that $\Xi$ is empty, (RC) reduces to a nominal optimization problem with no uncertainty:

$$ $\text{(P)}\quad\min_{x\in\mathcal{X}}f(x,a_{0}).$ $$

Consequently, we define the efficient decision, $x_{\mathrm{E}}$ as the optimizer to problem (P) and the robust decision, $x_{\mathrm{R}}$ as the optimizer to problem (RC), i.e.,

$$ $x_{\mathrm{E}}:=\arg\min_{x\in\mathcal{X}}\ f(x,a_{0}),\qquad x_{\mathrm{R}}:= \arg\min_{x\in\mathcal{X}}\max_{a\in\mathcal{U}}\ f(x,a).$ $$

For any $x\in\mathcal{X}$, its efficiency, $\mathrm{E}(x)$ and robustness, $\mathrm{R}(x)$ is defined respectively as its performance under nominal or worst-case uncertainty defined as

$$ $\mathrm{E}(x):=-f(x,a_{0}),\qquad\mathrm{R}(x):=-\max_{a\in\mathcal{U}}f(x,a).$ $$

It is easy to verify that the following inequalities hold

$$ $\mathrm{E}(x_{\mathrm{E}})\geq\mathrm{E}(x_{\mathrm{R}}),\qquad\mathrm{R}(x_{ \mathrm{R}})\geq\mathrm{R}(x_{\mathrm{E}}).$ $$

In practice, the trade-off between the efficiency and robustness of robust solutions can be controlled by adjusting the size of the uncertainty set. Specifically, we define an efficiency-robustness Pareto efficient robust solution with a nonempty compact convex uncertainty set $\mathcal{V}$ such that $a_{0}\in\mathcal{V}\subset\mathcal{U}$ as

$$ $x_{\mathrm{PE}}(\mathcal{V}):=\arg\min_{x\in\mathcal{X}}\max_{a\in\mathcal{V}} f(x,a).$ $$

For instance, under norm-ball uncertainty sets, we define the set of efficiency-robustness Pareto efficient robust solutions as the set of robust solutions generated while adjusting the radius of the norm-ball uncertainty set:

$$ $\left\{x_{\mathrm{PE}}(r):=\arg\min_{x\in\mathcal{X}}\max_{\xi\in\mathbb{B}_{ \|\cdot\|}(r)}f(x,a_{0}+\xi):\ r\in(0,\infty)\right\}.$ $$

Comparing with existing notions of Pareto efficiency of robust solutions (Iancu and Trichakis ()) which requires no other robust solutions that perform as least as well across all uncertainty realizations and better for some uncertainty realizations, we relax the requirement and trade-off worst-case performance with the performance under a nominal uncertainty realization.

### 2.3 Main Idea

If the uncertainty set radius $r$ for $\Xi$ is set too large, then a solution $x_{\mathrm{PE}}(r)$ of (RC) tends to be overly conservative and gives bad average case performance, i.e., low $\mathrm{E}(x_{\mathrm{PE}}(r))$. If the radius is set too small, then the solution may not perform reliably under large out-of-sample perturbations or adversarial attacks $\xi$, i.e., resulting in low $\mathrm{R}(x_{\mathrm{PE}}(r))$. What is the right $r$?

Existing literature (Mohajerin Esfahani and Kuhn (); Blanchet et al. (; )) points to ways to choose $r$ based on statistical theory, using observed perturbation data $\xi$ to estimate $\Xi$. But in practice, we may not have any data about $\Xi$. Even if we do, such designs are known in the robust optimization community to be overly conservative. The only practical option is to solve the problem (RC) many times, each with a different $r$, and compare the solutions in terms of their average case and worst-case performances.

Our main contribution is proposing a completely new way of thinking about this problem. Instead of solving (RC) $N$ times where $N$ is the number of solutions we want to compare, we can approximate these Pareto efficient robust solutions within two algorithmic passes. The first pass is to obtain a robust solution $x_{\mathrm{PE}}(r_{\mathrm{max}})$ for (RC) with a large value $r=r_{\mathrm{max}}$. Then we use $x_{\mathrm{PE}}(r_{\mathrm{max}})$ as the initial point of an iterative (proximal point) algorithm, to solve for the problem (RC) with $r=0$ (i.e., the nominal problem (P)). The intermediate iterates for $x$ provide a reasonable, sometimes perfect, representation of the Pareto efficient robust solutions.

## 3 Technical Preliminaries

### 3.1 Generalized Proximal Point Methods and The Central Paths

Here we introduce the generalized proximal point method and the central path for the nominal problem (P). Importantly, we introduce a preliminary result toward the proof of our main theorem, which is the equivalence between the sequence generated by the proximal point method for a linear problem (P) and the central path of the linear problem (P).

We begin with the introduction of the generalized proximal point method where a Bregman distance is in place of the usual Euclidean distance. First, we define the Bregman distance. We assume the distance-generating function $\varphi:\mathcal{X}\rightarrow\mathbb{R}$ satisfies the following technical assumptions:

- A1.
$\varphi$ is strictly convex, closed, and continuously differentiable in $\mathcal{X}$.
- A2.
If $\{x_{k}\}$ is a sequence in $\mathcal{X}$ which converges to a point $\overline{x}$ in the boundary of $\mathcal{X}$, and $y$ is any point in $\mathcal{X}$, then $\lim_{k\rightarrow\infty}\langle\nabla h(x_{k}),y-x_{k}=-\infty\rangle$.

The Bregman distance $D_{\varphi}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ w.r.t. $\varphi$ is defined as

$$ $D_{\varphi}(x,y)=\varphi(x)-\varphi(y)-\langle\nabla\varphi(y),x-y\rangle.$ $$

###### Example 1 .

- •
Let $\varphi(x)=\|x\|_{2}^{2}$, then $D_{\varphi}(x,y)=\|x-y\|_{2}^{2}$.
- •
Let $\varphi(x)=\langle x,\Sigma x\rangle$, then $D_{\varphi}(x,y)=\langle x-y,\Sigma(x-y)\rangle$.

Let $x_{0}\in\mathcal{X}$, the generalized proximal point method for solving the problem (P) generates a sequence $\{x_{k}\}\in\mathcal{X}$ as

$$ $x_{k+1}=\arg\min_{x\in\mathcal{X}}f(x,a_{0})+\lambda_{k}D_{\varphi}(x,x_{k})$ $$

where $\{\lambda_{k}\}\in\mathbb{R}_{++}$ satisfies
$\sum_{k=0}^{\infty}\lambda_{k}^{-1}=+\infty$.

The central path of the problem (P) with barrier function induced by the Bregman distance as $D_{\varphi}(\cdot,x_{0})$ is defined as $\{x(\omega):\omega\in(0,\infty)\}$ and

$$ $x(\omega)=\arg\min_{x\in\mathcal{X}}f(x,a_{0})+\omega D_{\varphi}(x,x_{0}).$ $$

The following result states for a linear problem (P), the proximal point method sequence and the central path are equivalent.

###### Proposition 1 ( Theorem 3 in Iusem et al. ( 1999 ) ) .

For problem (P) with linear objective functions $f(x,a_{0}):=\langle a_{0},x\rangle$ and a closed and convex polyhedron domain, $\mathcal{X}:=\left\{x\in\mathbb{R}^{n}:Ax\leq b\right\}$. Assume that $\varphi$ satisfies A1 and A2. Let $\{x(\omega):\omega\in(0,\infty)\}$ by the central path of the problem (P) w.r.t. $D_{\varphi}(\cdot,x_{0})$ and let $\{x_{k}\}$ be the proximal point method sequence for the problem (P) with Bregman distance $D_{\varphi}$. If the sequence ${\omega_{k}}$ is defined as

$$ $\omega_{k}=\left(\sum_{j=0}^{k-1}\lambda_{j}^{-1}\right)^{-1},\quad\text{for} \ k=1,2,...,$ $$

then

$$ $x_{k}=x(\omega_{k}),\quad\text{for}\ k=1,2,....$ $$

###### Example 1 .

###### Proposition 1 ( Theorem 3 in Iusem et al. ( 1999 ) ) .

### 3.2 Correspondence between Robust Optimization and Risk Measure Minimization

Our analysis also draws from the deep connection between robust optimization and risk measure minimization. Specifically, it has been shown that risk measures can be mapped explicitly to robust optimization uncertainty sets and vice versa (Natarajan et al. (); Bertsimas and Brown ()). In particular, we utilize the following correspondence between the robust optimization ellipsoidal uncertainty set and the mean-standard deviation risk measure.

###### Lemma 1 ( Correspondence between Ellipsoidal Robust Linear Optimization and and Mean-Variance Risk Minimization ) .

Under some closed convex polyhedron domain $\mathcal{X}\in\mathbb{R}^{n}$ and ellipsoidal uncertainty set $\Xi(\alpha)=\left\{\xi\in\mathbb{R}^{n}:\|\Sigma^{-1/2}\xi\|_{2}\leq\alpha\right\}$ where $\Sigma\in\mathbb{R}^{n\times n}$ is a symmetric matrix,
The robust optimization problem

$$ $\min_{x\in\mathcal{X}}\max_{\xi\in\Xi(\alpha)}\langle a_{0}+\xi,x\rangle$ (2) $$

is equivalent to the following mean-standard deviation risk measure minimization problem

$$ $\min_{x\in\mathcal{X}}\langle a_{0},x\rangle+\alpha\sqrt{\langle x,\Sigma x \rangle}.$ (3) $$

The proof of Lemma 3 is presented in Appendix A.

###### Lemma 1 ( Correspondence between Ellipsoidal Robust Linear Optimization and and Mean-Variance Risk Minimization ) .

## 4 Theory: Equivalence between the Pareto efficient Robust Solutions and the Proximal Point Method Trajectory

In this section, we show that under the simplex domain and ellipsoidal uncertainty set, a set of Pareto efficient robust solutions to uncertain linear optimization problems with uncertainty in the objective function can be obtained exactly as a proximal point method trajectory. Further, for random problem instances with random polyhedron domains, we show that the performances of the Pareto efficient solutions are bounded probabilistically in between the performances of two proximal point method trajectories. The main implication of our result is the following: instead of solving a different instance of a robust optimization problem to arrive at each solution on the efficiency-robustness Pareto frontier, an entire “menu” of solutions on the efficiency-robustness Pareto frontier can be obtained approximately, under some condition exactly in a single pass via gradient methods.

### 4.1 Exact Result: Simplex Domain and Ellipsoidal Uncertainty Set

We first consider the case of Pareto efficient robust solutions to robust linear optimization problems with uncertain objective functions under the simplex domain and ellipsoidal uncertainty sets. We show a proximal point sequence is contained within the set of Pareto efficient robust solutions. In particular, we have the following theorem.

###### Theorem 1 ( Correspondence Between Pareto-Efficiency Robust Solutions and the Proximal Point Trajectory ) .

Under linear objective functions, $f(x,a):=\langle a,x\rangle$. Let $\left\{x_{\mathrm{PE}}(\alpha):\alpha>0\right\}$ be the set of Pareto efficient robust solutions under simplex domain $\Delta^{n}=\left\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle=1\right\}$ and ellipsoidal uncertainty set $\Xi(\alpha)=\left\{\xi\in\mathbb{R}^{n}:\|\Sigma^{-1/2}\xi\|_{2}\leq\alpha\right\}$ where $\Sigma$ satisfies $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$. Let $\{x_{k}\}$ be the proximal point sequence w.r.t. $D_{\varphi}(x,y)=\langle x-y,\Sigma(x-y)\rangle$, associated with sequence $\{\lambda_{k}\}$ and starting point $x_{\mathrm{R}}=\arg\min_{x\in\mathcal{X}}\max_{\xi\in\Xi(\infty)}\langle a_{0}
+\xi,x\rangle$. If the sequence $\{\omega_{k}\}$ is defined as

$$ $\omega_{k}=\left(\sum_{j=0}^{k-1}\lambda_{j}^{-1}\right)^{-1},\quad\text{for} \ k=1,2,...,$ $$

and let $\alpha(\omega_{k})$ be such that $\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\alpha(\omega_{k})\sqrt{
\langle x,\Sigma x\rangle}=\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+
\omega_{k}\langle x,\Sigma x\rangle$.
Then

$$ $x_{k}=x_{\mathrm{PE}}(\alpha(\omega_{k})),\quad\text{for}\ k=1,2,....$ $$

###### Theorem 1 ( Correspondence Between Pareto-Efficiency Robust Solutions and the Proximal Point Trajectory ) .

### 4.2 Implication of Theorem 1

Theorem 1 inspires an efficient algorithm for generating multiple (approximate) Pareto efficient robust solutions in two passes: first, solve the robust problem (RC) for $x_{\mathrm{R}}$; second, solve the nominal problem (P) with proximal point method initialized with $x_{\mathrm{R}}$, finally the trajectory of the proximal point method are approximate Pareto efficient robust solutions. Specifically, we present the following algorithms.

Figure: Algorithm 1 Multiple Approximate Pareto efficient Robust Solutions via Proximal Point Methods

For linear objective functions, the proximal point method updates are reduced to vanilla gradient descent updates. For general objective functions beyond linear functions, performing the proximal point method may not be computationally affordable. In practice, the proximal point method can be approximated by computationally cheap extra-gradient and optimistic gradient methods (Mokhtari et al. ()). In Section 5, we demonstrate the empirical performance of Algorithm 1 for general problems where the conditions for our exact results in Theorem 1 no longer hold.

### 4.3 Proof of Theorem 1

Theorem 1 follows by the combination of Proposition 1, 2 and 3, which together show the notions of proximal-point sequences, central path, mean-variance risk minimization solutions and Pareto efficient robust solutions all coincide under simplex domain and ellipsoidal uncertainty set. Here we present Proposition 2 and 3.

###### Proposition 2 ( Correspondence between Mean-Variance Risk Minimization Solutions and the Central Path ) .

Under simplex domain $\Delta^{n}:=\left\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle=1\right\}$, assume the covariance matrix of the returns $\Sigma$ satisfies $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$, Let the set of Pareto efficient solutions to the mean-variance risk measure minimization problem be

$$ $\left\{x_{\mathrm{RM}}(\alpha)=\arg\min\left\{\langle a_{0},x\rangle+\alpha \langle x,\Sigma x\rangle:\ x\in\Delta^{n}\right\}:\ \alpha\in(0,\infty)\right\}.$ $$

Let the central path be

$$ $\left\{x_{\mathrm{CP}}(\omega)=\arg\min\left\{\langle a_{0},x\rangle+\omega D_ {\varphi,\Sigma}(x,x_{\mathrm{mv}}):\ x\in\Delta^{n}\right\}:\ \omega\in(0, \infty)\right\},$ $$

where $D_{\varphi,\Sigma}(x,x_{\mathrm{mv}})=\langle x-x_{\mathrm{mv}},\Sigma(x-x_{
\mathrm{mv}})\rangle$ which is induced by $\varphi(x)=\langle x,\Sigma x\rangle$ and $x_{\mathrm{mv}}$ is the minimum variance solution defined as $x_{\mathrm{mv}}=\arg\min\left\{\langle x,\Sigma x\rangle:\ x\in\Delta^{n}\right\}$.
Then

$$ $x_{\mathrm{RM}}(\alpha)=x_{\mathrm{CP}}(\alpha),\quad\forall\alpha\in(0,\infty).$ $$

The proof of Proposition 2 is presented in Appendix B.

###### Proposition 3 ( Correspondence Between Pareto-Efficiency Robust Solutions and the Central Path ) .

Under simplex domain $\Delta^{n}=\left\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle=1\right\}$ and ellipsoidal uncertainty set $\Xi(\alpha)=\left\{\xi\in\mathbb{R}^{n}:\|\Sigma^{-1/2}\xi\|_{2}\leq\alpha\right\}$ where $\Sigma$ satisfies $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$, let the Pareto efficient robust solutions be

$$ $\left\{x_{\mathrm{PE}}(\alpha)=\arg\min_{x\in\Delta^{n}}\max_{\xi\in\Xi(\alpha )}\langle a_{0}+\xi,x\rangle:\ \alpha\in(0,\infty)\right\}.$ $$

Let the central path be

$$ $\left\{x_{\mathrm{CP}}(\omega)=\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle +\omega D_{\varphi,\Sigma}(x,x_{\mathrm{R}}):\ \omega\in(0,\infty)\right\}$ $$

where $D_{\varphi,\Sigma}(x,x_{\mathrm{R}})=\langle x-x_{\mathrm{R}},\Sigma(x-x_{
\mathrm{R}})\rangle$ which is induced by $\varphi(x)=\langle x,\Sigma x\rangle$ and $x_{\mathrm{R}}=\arg\min_{x\in\mathcal{X}}\max_{\xi\in\Xi(\infty)}\langle a_{0}
+\xi,x\rangle$. Let $\alpha(\omega)$ be such that $\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\alpha(\omega)\sqrt{\langle x,
\Sigma x\rangle}=\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\omega\langle
x
,\Sigma x\rangle$. Then

$$ $x_{\mathrm{PE}}(\alpha(\omega))=x_{\mathrm{CP}}(\omega),\quad\forall\alpha\in( 0,\infty).$ $$

The proof of Proposition 3 is presented in Appendix B.

###### Proposition 2 ( Correspondence between Mean-Variance Risk Minimization Solutions and the Central Path ) .

###### Proposition 3 ( Correspondence Between Pareto-Efficiency Robust Solutions and the Central Path ) .

### 4.4 Probabilistic Performance Bound: Random Polyhedron Domain and Ellipsoidal Uncertainty Set

To extend the result to more general domains beyond simplex domains. In this subsection, we show that under ellipsoidal uncertainty sets, the performance (measured by efficiency and robustness) of the Pareto efficient robust solutions with random polyhedron domains with constraint coefficients generated i.i.d. from bounded distributions are bounded between the performance of two sets of Pareto efficient robust solutions with two simplex domains with a small scaling factor, hence by Theorem 1, between the performance of two proximal point trajectories with high probability.

We denote the Pareto efficient robust solution with domain $\mathcal{S}$ and ellipsoidal uncertainty set $\Xi(\alpha)=\left\{\xi\in\mathbb{R}^{n}:\|\Sigma^{-1/2}\xi\|_{2}\leq\alpha\right\}$ as

$$ $\displaystyle x_{\mathrm{PE}}(\alpha,\mathcal{S})=$ $\displaystyle\arg\min_{x\in\mathcal{S}}\max_{\xi\in\Xi(\alpha)}\langle a_{0}+ \xi,x\rangle,$ $\displaystyle=$ $\displaystyle\arg\min_{x\in\mathcal{S}}\langle a_{0},x\rangle+\alpha\sqrt{ \langle x,\Sigma x\rangle}.$ $$

Alternatively, we define

$$ $\displaystyle x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha),\mathcal{S})=$ $\displaystyle\arg\min\left\{\sqrt{\langle x,\Sigma x\rangle}:\langle a_{0},x \rangle\leq\upsilon(\alpha),\ x\in\mathcal{S}\right\},$ $$

where we assume $\upsilon(\alpha)$ is chosen such that $x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha),\mathcal{S})=x_{\mathrm{PE}}(\alpha,
\mathcal{S})$.

###### Corollary 4 .

Consider a random polyhedron $\tilde{\mathcal{X}}=\{x\in\mathbb{R}^{n}_{+}:\tilde{\mathbf{A}}x\leq\overline{
d}\cdot e\}$, where $\tilde{\mathbf{A}}\in\mathbb{R}^{m\times n}$ and $\tilde{A}_{ij}$ are i.i.d. according to a bounded distribution with support $[0,b]$ and $\mathbb{E}[\tilde{A}_{ij}]=\mu$ for all $i\in[m],j\in[n]$. Define simplex $\Delta=\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle\leq\frac{\overline{d}}{b}\}$. Then for all $\alpha\in(0,\infty)$,

$$ $\mathbb{P}\left(\mathrm{R}(x_{\mathrm{PE}}(\alpha,\Delta))\leq\mathrm{R}(x_{ \mathrm{PE}}(\alpha,\tilde{\mathcal{X}}))\leq\mathrm{R}(x_{\mathrm{PE}}(\alpha ,\frac{b}{\mu(1-\epsilon)}\cdot\Delta))\right)=1-\frac{1}{m},$ $$

where $\epsilon=\frac{b}{\mu}\sqrt{\frac{\log m}{n}}$.

The proof of Corollary 4 is presented in Appendix B.

###### Corollary 4 .

### 4.5 Multiple Uncertain Constraints

In this subsection, we extend the result to robust linear optimization problems with multiple uncertain linear constraints. We show that under uncertain linear constraints, the Pareto efficient robust solutions can be obtained by solving a set of saddle point problems. Specifically, we study the following problem

$$ $\textrm{(RCWUC)}\quad\min\left\{\langle a_{0},x\rangle:\ \sup_{\xi_{i}\in\Xi_{ i}}\langle a_{i}+\xi_{i},x\rangle-b_{i}\leq 0,\ \forall i\in[m],\ x\in\Delta^{ n}\right\},$ $$

where $\Xi_{i}$ is the uncertainty set for uncertain parameter $\xi_{i}$ in constraint $i$ for all $i\in[m]$. We assume the constraints share the same ellipsoidal uncertainty set, i.e., $\Xi_{i}(r)=\{\xi\in\mathbb{R}^{n}:\|\Sigma^{-1/2}\xi\|_{2}\leq r\},\ \forall i
\in[m]$. Consequently, the Pareto efficient robust solutions are the set

$$ $\{x_{\mathrm{PE}}^{\prime}(\beta):\ \beta\in[0,\infty)\},$ $$

where

$$ $x_{\mathrm{PE}}^{\prime}(\beta)=\arg\min\left\{\langle c_{0},x\rangle:\ \sup_{ \xi_{i}\in\Xi_{i}(\beta)}\langle c_{i}+\xi_{i},x\rangle-b_{i}\leq 0,\ \forall i \in[m],\ x\in\Delta^{n}\right\}.$ $$

In general, the conventional approach for solving for a Pareto efficient robust solution requires dual reformulation of the robust constraints resulting in a computationally harder problem than the deterministic counterpart. Our next result shows for the specific problem setting of (RCWUC), its Pareto efficient robust solutions can be obtained by solving a set of saddle-point problems via gradient descent-ascent-like methods.

###### Proposition 5 .

$\{x_{\mathrm{PE}}^{\prime}(\beta):\ \beta\in[0,\infty)\}$ is equal to a set of saddle-point solutions $\{x_{\mathrm{SP}}(\alpha):\ \alpha\in[0,\infty)\}$ where

$$ $x_{\mathrm{SP}}(\alpha)=\arg\min_{x\in\Delta^{n}}\left\{\max_{\lambda\in \mathbb{R}^{m}_{+}}\langle c_{0},x\rangle+\langle\lambda,Cx\rangle+\alpha \langle\lambda,e\rangle\langle x,\Sigma x\rangle-\langle\lambda,b\rangle\right\}.$ $$

The proof of Proposition 5 is presented in Appendix B. As a result of Proposition 5, a set of approximate Pareto efficient robust solutions can be obtained by running the following algorithm for a set of $\alpha$ values.

Figure: Algorithm 2 $\hat{x}_{\mathrm{SP}}(\alpha)$ Oracle

###### Proposition 5 .

## 5 Performance of Algorithm 1 : Numerical Studies

In this section, we present two experiments testing the empirical performance of generating a set of approximate efficiency-robustness Pareto efficient robust solutions via Algorithm 1.

### 5.1 Robust Portfolio Optimization

Figure: Figure 1: Robust Portfolio Optimization: The first two figures compare the in-sample and the out-of-sample performances (as measured by the nominal and worst-case returns) of the exact Pareto efficient portfolios against our PPM trajectory approximated Pareto efficient portfolios for the basic robust portfolio problem (RPO); the last two figures are from the same experiment but on the Markowitz++ problem. The results show Algorithm 1 generates closed approximations of Pareto efficient portfolios measured by both the in-sample and out-of-sample performances.
Refer to caption: x1.png

Robust Portfolio Optimization. In portfolio optimization, we are concerned with constructing a portfolio as a convex combination of $n$ stocks. The returns of the $n$ stocks are modeled by a random vector, $r$. We assume that from historical data, we estimated the expectation and the covariance matrix of $r$ to be $\mu$ and $\Sigma$. In the robust approach to the problem, we assume the realization of the uncertain return lies within an uncertainty set which we design as an ellipsoidal uncertainty set $\mathcal{U}(\alpha)=\{\mu+\xi\in\mathbb{R}^{n}:\ \|\Sigma^{-1/2}\xi\|_{2}\leq\alpha\}$. Our objective is to choose the portfolio weight that minimizes the worst-case loss (or equivalently maximizes the worst-case return) under a simplex domain, leading to the following robust portfolio optimization problem,

$$ $\displaystyle\text{(RPO)}\quad\min_{x\in{\Delta^{n}}}\max_{r\in\mathcal{U}( \alpha)}-\langle r,x\rangle.$ $$

As a consequence of Lemma 3, (RPO) is equivalent to the classic mean-standard deviation risk measure minimization problem,

$$ $\displaystyle\text{(RMM)}\quad\min_{x\in{\Delta^{n}}}-\langle\mu,x\rangle+ \alpha\sqrt{\langle x,\Sigma x\rangle}.$ $$

To further test the performance of algorithm 1 beyond the simplex domain required for our exact result in Theorem 1, we also consider the extended Markowitz model (Markowitz++) proposed by Boyd et al. () with additional practical constraints and objective terms corresponding to e.g., shorting, weight limits, cash holding/borrowing constraints and costs. The detailed formulation of the Markowitz++ portfolio problem is in Appendix C.

Experiment design. We first construct portfolios with in-sample historical stock return data, before testing the nominal and worst-case returns of each portfolio on out-of-sample stock return data. As the benchmark, we first construct exact Pareto efficient robust portfolios by solving exactly the problem (RPO)/Markowitz++ under various levels of $\alpha$. Then we deploy algorithm 1 to generate approximate Pareto efficient robust portfolios, i.e., we use the (RPO)/Markowitz++ portfolio with the largest $\alpha$ to initialize the PPM method for solving the nominal portfolio optimization problem with $\alpha=0$, the PPM trajectory are approximate Pareto efficient robust portfolios. Specifically, we choose our decision space as $20$ stocks from S$\&$P 500 companies. We use the historical daily return data of the 20 stocks over 3 years (from 2021-01-01 to 2023-12-30) and estimate the moment information $\mu_{\mathrm{in}}$ and $\Sigma_{\mathrm{in}}$ as our in-sample data for constructing portfolios; the assumed unseen future daily return of the 20 stock for the next 8 months (from 2024-01-01 to 2024-08-01) are used to estimate $\mu_{\mathrm{out}}$ and $\Sigma_{\mathrm{out}}$ which serve as our out-of-sample data for evaluating the nominal and worst-case returns of each portfolio.

Results. The result is shown in figure 1. For the vanilla robust portfolio problem (RPO), although the assumption $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$ in Theorem 1 is not satisfied in this experiment, the performance of PPM method generated portfolios matches closely to that of the exact Pareto efficient robust portfolios. For the Markowitz++ portfolio problem, the domain further deviates from our simplex requirement for the exact result in Theorem 1, algorithm 1 still produces good approximate Pareto efficient robust portfolios. At the same time, the computation cost of solving multiple (RPO) problems with various levels of $\alpha$ is reduced to solving the most robust (RPO) problem and solving the nominal problem by performing PPM updates.

### 5.2 Adversarially Robust Deep Learning

Figure: Figure 2: Adversarially Robust Deep Learning: Clean test accuracy and PGD adversarial test accuracy of Algorithm 1 generated approximate Pareto efficient robust networks v.s. adversarially trained Pareto efficient robust networks. Four variants of the gradient method are implemented in Algorithm 1 for standard training with robust parameter initialization. The first two figures show performance improvement by using the extra gradient, the last two figures show performance improvement by using the full gradient. One insight is that the performance of Algorithm 1 can be improved by improving the gradient method approximation to PPM during standard training. The best variant: Algorithm 1 with ExtraFullGD generated 100 approximations of Pareto efficient robust networks in two algorithmic passes.
Refer to caption: x2.png

**Table 1: Computation Cost: Algorithm 1 vs. Adversarial Training**
| Method | Cost per Pareto efficient robust network (min) | Cost to generate $N$ Pareto efficient robust networks (min) |
| --- | --- | --- |
| Algorithm 1 with ExtraFullGD | $0.25$ | $15.12+0.25(N-1)$ |
| FGSM (Wong et al. ()) | $15.12$ | $15.12N$ |

In this subsection, we present experiment results that evaluate the performance of our PPM-based procedure (Algorithm 1) for efficiently generating multiple adversarially robust neural networks trading-off clean accuracy (test accuracy with a clean test set) with adversarial accuracy (test accuracy with a test set under adversarial perturbations).

Adversarial training as robust optimization. The goal in adversarially robust deep learning is to learn networks that are robust against adversarial attacks (i.e., perturbations on the input examples that aim to deteriorate the accuracy of classifiers). A common strategy to robustify networks is adversarial training, which can be formulated as the following robust optimization problem,

$$ $\min_{\theta}\mathbb{E}_{(x,y)\sim\mathcal{D}}\left[\max_{\xi\in\Xi}\ell(f_{ \theta}(x+\xi),y)\right],$ (8) $$

where $\mathcal{D}$ is the distribution generating pairs of examples $x\in\mathbb{R}^{d}$ and labels $y\in[c]$, $f_{\theta}$ is a neural network parameterized by $\theta$, $\xi$ is the perturbation with perturbation set $\Xi$ , and $\ell$ is the lost function. Typically, the perturbation set is designed as norm-balls, $\mathbb{B}_{\|\cdot\|}(r)$ whose size can be controlled via the radius parameter, $r$. Naturally, we can apply our framework and define a clean accuracy-adversarial accuracy Pareto efficient robust networks as networks adversarially trained under different levels of perturbation set radius, $r$. The trade-off between clean accuracy and adversarial accuracy in adversarially robust deep learning has been observed empirically Wang et al. (); Su et al. () and studied theoretically Raghunathan et al. (); Tsipras et al. (); Pang et al. (). In practice, given the nonconvex-nonconcave loss function, the adversarial training is solved approximately via iteratively performing the following: first approximately solving the inner maximization problem, followed by gradient method update on the parameter $\theta$ (Madry et al. ()).

Experiment design. As the benchmark for algorithm 1, we first adversarially train networks under different perturbation set radius, $r$ to learn a set of clean accuracy-adversarial accuracy Pareto efficient robust networks. Then we run Algorithm 1 by performing standard training but with the key difference of initializing the network parameter with the parameter of the most robust network (i.e., the adversarially trained network with the largest $r$). Finally, the output of Algorithm 1, i.e., the standard training parameter sequence with robust parameter initialization, $\{\theta_{k}\}$ corresponds to a set of approximate clean accuracy-adversarial accuracy Pareto efficient robust networks.

Specifically, we use the CIFAR10 dataset, a PreAct ResNet18 architecture, and the cross entropy loss. The set of Pareto efficient robust networks are adversarially trained using fast gradient sign method (FGSM) with random initialization and fast training methods (Wong et al. ()) (shown to be as effective as projected gradient descent (PGD)-based training but with a much lower cost) and $l_{\infty}$ norm ball perturbation sets, $\mathbb{B}_{\|\cdot\|_{\infty}}(r)$, with $r$ in $\{2,4,6,8\}$. The Algorithm 1 is initialized with the parameter of the adversarially trained network with perturbation sets $\mathbb{B}_{\|\cdot\|_{\infty}}(r=8)$, after initialization, four variants of the standard training are performed with the following gradient methods: 1) Vanilla stochastic gradient descent (SGD); 2) Stochastic extra gradient descent (ExtraSGD); 3) Gradient descent with the gradient of the full train set (FullGD); 4) Extra gradient descent with the gradient of the full train set (ExtraFullGD). The implementation for extra gradient descent is adopted from Gidel et al. (). Each variant can be considered as an approximation to PPM in Algorithm 1, where ExtraSGD is a better approximation to PPM than SGD, and ExtraFullGD is a better approximation to PPM than ExtraSGD. SGD and ExtraSGD have a learning rate of $5\mathrm{e}-4$; FullGD and ExtraFullGD have a learning rate of $4\mathrm{e}-4$. The four variants of the standard training are each performed for 100 epochs, generating one approximate Pareto efficient robust network per epoch. The trajectories of four variants of the standard training are each a set of approximate clean accuracy-adversarial accuracy Pareto efficient robust networks. Finally, all networks’ clean accuracy and adversarial accuracy are evaluated correspondingly on a clean test set and on an adversarial test set with PGD attacks.

Results. The results evaluating the performance of our PPM-based procedure for generating approximate clean (test) accuracy-adversarial (test) accuracy Pareto efficient robust networks are shown in Figure 2. The networks generated by our gradient method trajectories initially approximate/surpass both the clean and adversarial accuracy of adversarially trained networks with perturbation set radius, $r$ in $\{8,6,4\}$, before generating networks with both lower clean and adversarial accuracy that than of adversarially trained network with $r=2$. The initial approximation/surpassing in performance against adversarially trained networks shows our procedure can generate adversarially robust networks as effective as traditional adversarial training. The later drop in performance is mainly contributed by the low learning rate, although tuning the gradient methods is not the focus of our paper, it opens up future works for investigating the gradient method designs in Algorithm 1 that can improve upon our current approximates using constant learning rates. Comparing the performance across the four trajectories corresponding to the four gradient method variants, using the full gradient of the training set improves both clean and adversarial accuracy against using the stochastic gradient; extra gradient improves both the clean and adversarial accuracy against the vanilla gradient, with the ExtraFullGD trajectory generated networks having the best performance amount the four sets of networks. This result is as expected since ExtraFullGD is the best approximation for PPM.

The computation cost result in Table 1 shows our PPM-based procedure reduces the computation cost of generating $N$ Pareto efficient robust networks from $N\times T$ to $2\times T$, where $T$ is the cost of a single adversarial training. Specifically, algorithm 1 generates a set of $N$ approximate Pareto efficient robust networks with the cost of a single pass of adversarial training (costing 15.12 minutes) plus a single pass of standard training with robust parameter initialization for $N-1$ epochs (costing $0.25(N-1)$ minutes, where each epoch of standard training costs $0.25$ minutes, and one approximate Pareto efficient robust network is generated per epoch).

## Appendix A Proofs of Results in Section 3

Proof of Lemma 3.
Denote $z=\Sigma^{-1/2}\xi$,

$$ $\displaystyle\min_{x\in\mathcal{X}}\max_{\xi\in\Xi(\alpha)}\langle a_{0}+\xi,x\rangle$ $\displaystyle=$ $\displaystyle\min_{x\in\mathcal{X}}\langle a_{0},x\rangle+\max_{\xi\in\Xi( \alpha)}\langle\xi,x\rangle$ $\displaystyle=$ $\displaystyle\min_{x\in\mathcal{X}}\langle a_{0},x\rangle+\max_{z\in\mathbb{B} _{\|\cdot\|_{2}}(\alpha)}\langle z,\Sigma^{1/2}x\rangle$ $\displaystyle=$ $\displaystyle\min_{x\in\mathcal{X}}\langle a_{0},x\rangle+\alpha\sqrt{\langle x ,\Sigma x\rangle}.$ $$

## Appendix B Proofs of Results in Section 4

###### Lemma 2 .

Assume $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$,

$$ $\arg\min\left\{\langle x,\Sigma x\rangle:\ x\in\Delta^{n}\right\}=\frac{\Sigma ^{-1}e}{\langle e,\Sigma^{-1}e\rangle}.$ $$

Proof.
By KKT condition,

$$ $\arg\min\left\{\langle x,\Sigma x\rangle:\ \langle e,x\rangle=1\right\}=\frac{ \Sigma^{-1}e}{\langle e,\Sigma^{-1}e\rangle},$ $$

denote $\overline{x}=\frac{\Sigma^{-1}e}{\langle e,\Sigma^{-1}e\rangle}$ we have,

$$ $\langle\overline{x},\Sigma\overline{x}\rangle\leq\langle x,\Sigma x\rangle, \quad\forall x\in\{x\in\mathbb{R}^{n}:\langle e,x\rangle=1\}.$ $$

Given $\Delta^{n}:=\left\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle=1\right\}\subset
\{x\in\mathbb{R}^{n}:\langle e,x\rangle=1\}$, we have

$$ $\langle\overline{x},\Sigma\overline{x}\rangle\leq\langle x,\Sigma x\rangle, \quad\forall x\in\Delta^{n}.$ $$

Finally, by $\Sigma^{-1}e\in\mathbb{R}^{n}_{+}$,

$$ $\overline{x}\in\Delta^{n},$ $$

hence, $\arg\min\left\{\langle x,\Sigma x\rangle:\ x\in\Delta^{n}\right\}=\frac{\Sigma
^{-1}e}{\langle e,\Sigma^{-1}e\rangle}$.

Proof of Proposition 2.
For all $\alpha\in(0,\infty)$ we have

$$ $\displaystyle x_{\mathrm{CP}}(\alpha)=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha D_{\varphi,\Sigma}(x ,x_{\mathrm{mv}}):\ x\in\Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha\langle x-x_{\mathrm{ mv}},\Sigma(x-x_{\mathrm{mv}})\rangle:\ x\in\Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha\langle x,\Sigma x \rangle-2\alpha\langle x_{\mathrm{mv}},\Sigma x\rangle:\ x\in\Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha\langle x,\Sigma x \rangle-\frac{2\alpha}{\langle e,\Sigma^{-1}e\rangle}\langle e,x\rangle:\ x\in \Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha\langle x,\Sigma x \rangle-\frac{2\alpha}{\langle e,\Sigma^{-1}e\rangle}:\ x\in\Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle\arg\min\left\{\langle a_{0},x\rangle+\alpha\langle x,\Sigma x \rangle:\ x\in\Delta^{n}\right\}$ $\displaystyle=$ $\displaystyle x_{\mathrm{RM}}(\alpha),$ $$

where the fourth equality is by Lemma 2, and the fifth equality is due to $x\in\Delta^{n}$.

Proof of Proposition 3. The result is a direct consequence of Lemma 3 and Proposition 2.

$$ $\displaystyle x_{\mathrm{PE}}(\alpha(\omega))=$ $\displaystyle\arg\min_{x\in\Delta^{n}}\max_{\xi\in\Xi(\alpha(\omega))}\langle a _{0}+\xi,x\rangle$ $\displaystyle=$ $\displaystyle\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\alpha(\omega) \sqrt{\langle x,\Sigma x\rangle}$ $\displaystyle=$ $\displaystyle\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\omega\langle x, \Sigma x\rangle$ $\displaystyle=$ $\displaystyle\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\omega D_{\varphi ,\Sigma}(x,x_{\mathrm{MV}})$ $\displaystyle=$ $\displaystyle\arg\min_{x\in\Delta^{n}}\langle a_{0},x\rangle+\omega D_{\varphi ,\Sigma}(x,x_{\mathrm{R}})$ $\displaystyle=$ $\displaystyle x_{\mathrm{CP}}(\omega).$ $$

The second equality is due to Lemma 3, the fourth equality is by Proposition 2, and finally the fifth equality follows from Lemma 3.

Our performance bound in corollary 4. builds on the following result that a random polyhedron with constraint coefficients generated i.i.d. according to bounded distributions is sandwiched between two simplices with high probability.

###### Lemma 3 ( Theorem 2.1. in El Housni and Goyal [ 2017 ] ) .

Consider a random polyhedron $\tilde{\mathcal{X}}=\{x\in\mathbb{R}^{n}_{+}:\tilde{\mathbf{A}}x\leq\overline{
d}\cdot e\}$, where $\tilde{\mathbf{A}}\in\mathbb{R}^{m\times n}$ and $\tilde{A}_{ij}$ are i.i.d. according to a bounded distribution with support $[0,b]$ and $\mathbb{E}[\tilde{A}_{ij}]=\mu$ for all $i\in[m],j\in[n]$. Define simplex $\Delta=\{x\in\mathbb{R}^{n}_{+}:\langle e,x\rangle\leq\frac{\overline{d}}{b}\}$. Then

$$ $\mathbb{P}\left(\Delta\subseteq\tilde{\mathcal{X}}\subseteq\frac{b}{\mu(1- \epsilon)}\cdot\Delta\right)=1-\frac{1}{m},$ $$

where $\epsilon=\frac{b}{\mu}\sqrt{\frac{\log m}{n}}$.

Proof of Corollary 4.
Denote $\kappa=\frac{b}{\mu(1-\epsilon)}$ where $\epsilon=\frac{b}{\mu}\sqrt{\frac{\log m}{n}}$.

$$ $\displaystyle\mathbb{P}\left(\Delta\subseteq\tilde{\mathcal{X}}\subseteq\kappa \cdot\Delta\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\min_{x:\langle a_{0},x\rangle\leq\upsilon(\alpha ),\ x\in\kappa\cdot\Delta}\sqrt{\langle x,\Sigma x\rangle}\leq\min_{x:\langle a _{0},x\rangle\leq\upsilon(\alpha),\ x\in\tilde{\mathcal{X}}}\sqrt{\langle x, \Sigma x\rangle}\leq\min_{x:\langle a_{0},x\rangle\leq\upsilon(\alpha),\ x\in \Delta}\sqrt{\langle x,\Sigma x\rangle}\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\sqrt{\langle x,\Sigma x\rangle}|x=x_{\mathrm{PE} }^{\prime}(\upsilon(\alpha),\kappa\cdot\Delta)\leq\sqrt{\langle x,\Sigma x \rangle}|x=x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha),\tilde{\mathcal{X}})\leq \sqrt{\langle x,\Sigma x\rangle}|x=x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha), \cdot\Delta)\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\mathrm{R}(x_{\mathrm{PE}}^{\prime}(\upsilon( \alpha),\Delta))\leq\mathrm{R}(x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha), \tilde{\mathcal{X}}))\leq\mathrm{R}(x_{\mathrm{PE}}^{\prime}(\upsilon(\alpha), \kappa\cdot\Delta))\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\mathrm{R}(x_{\mathrm{PE}}(\alpha,\Delta))\leq \mathrm{R}(x_{\mathrm{PE}}(\alpha,\tilde{\mathcal{X}}))\leq\mathrm{R}(x_{ \mathrm{PE}}(\alpha,\kappa\cdot\Delta))\right).$ $$

By Lemma 3 we have result.

Proof of Proposition 5.
By Proposition 3,

$$ $x_{\mathrm{PE}}^{\prime}(\beta)=\arg\min\left\{\langle c_{0},x\rangle:\ \langle c_{i},x\rangle+\beta\sqrt{\langle x,\Sigma x\rangle}-b_{i}\leq 0,\ \forall i\in[m],\ x\in\Delta^{n}\right\}.$ $$

By the monotonicity of $\sqrt{\ \cdot\ }$,

$$ $\{x_{\mathrm{SP}}(\alpha):\ \alpha\in[0,\infty)\}=\{x_{\mathrm{PE}}^{\prime}( \beta):\ \beta\in[0,\infty)\},$ $$

where

$$ $x_{\mathrm{SP}}(\alpha)=\arg\min\left\{\langle c_{0},x\rangle:\ \langle c_{i}, x\rangle+\alpha\langle x,\Sigma x\rangle-b_{i}\leq 0,\ \forall i\in[m],\ x\in \Delta^{n}\right\}.$ $$

Applying the method of Lagrangian, $x_{\mathrm{SP}}(\alpha)$ can reformulated as the solution to the following saddle-point problem

$$ $\displaystyle x_{\mathrm{SP}}(\alpha)$ $\displaystyle=\arg\min_{x\in\Delta^{n}}\left\{\max_{\lambda\in\mathbb{R}^{m}_{ +}}\langle c_{0},x\rangle+\sum_{i\in[m]}\lambda_{i}\left(\langle c_{i},x \rangle+\alpha\langle x,\Sigma x\rangle-b_{i}\right)\right\}$ $\displaystyle=\arg\min_{x\in\Delta^{n}}\left\{\max_{\lambda\in\mathbb{R}^{m}_{ +}}\langle c_{0},x\rangle+\langle\lambda,Cx\rangle+\alpha\langle\lambda,e \rangle\langle x,\Sigma x\rangle-\langle\lambda,b\rangle\right\}.$ $$

###### Lemma 2 .

###### Lemma 3 ( Theorem 2.1. in El Housni and Goyal [ 2017 ] ) .

## Appendix C Markowitz++

The Markowitz++ proposed by Boyd et al. [] extends the classical Markowitz Portfolio by introducing practical constraints and objective terms. In the experiment, we include the option for shorting, holding limits on each asset, the option for cash holding/borrowing, and the associated costs. The formulation of the Markowitz++ problem is shown below,

$$ $\displaystyle\min\quad$ $\displaystyle\ -\langle\mu,x\rangle+\gamma^{\text{hold}}\phi^{\text{hold}}(x,c )+\alpha\sqrt{\langle x,\Sigma x\rangle}$ (14a) s.t. $\displaystyle\langle e,x\rangle+c=1,$ (14b) $\displaystyle x^{\text{min}}\leq x\leq x^{\text{max}},$ (14c) $\displaystyle c^{\text{min}}\leq c\leq c^{\text{max}}.$ (14d) $$

The decision variables $x\in\mathbb{R}^{n}$ are stock weights ($x_{i}>0$ for holding, $x_{i}<0$ for shorting), $c\in\mathbb{R}$ is the cash weight ($c>0$ denotes diluting the portfolio with cash, $c<0$ denotes borrowing). The objective function now includes the additional term of holding costs, $\phi^{\text{hold}}(x,c)$ as defined by equation (5) in Boyd et al. [] and weighted by the parameter $\gamma^{\text{hold}}$. The first constraint enforces all weights sum to one, the second constraint corresponds to the maximum shorting position and maximum holding position on each asset, and the last constraint applies lower and upper limits on the cash weight. Finally, the nominal and worst-case return trade-off is controlled by adjusting the parameter, $\alpha$.

## Peer Reviews

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: See above.
soundness: 2
strengths: This paper proposes a novel way for solving robust optimization with an uncertainty set. The idea of applying Proximal point method for computing the Pareto efficient robust solutions seems to be a novel idea that has been explored before. 

Empirical results demonstrate the effectiveness of the proposed methods.
summary: This paper studies robust optimization with an uncertainty set, and proposes a more efficient way for computing Pareto efficient robust solutions based on proximal point methods.
weaknesses: A limited setting: The algorithms and theoretical guarantees given by the authors seem to only work on a toy example with linear functions and ellipsoidal uncertainty set. The objective function (e.g., in (3)) has a very particular form, and it seems that the analysis is difficult to be generalized to other cases.

One of the main advantage mentioned by the authors is that the total computational cost is reduced from NT to 2T. I wonder can the authors provide more discussion on this point? To be more specific, which papers are we talking here (Line 054)? 

I am not sure how to understand Theorem 1. The algorithm gives a series of x_k, which are PE in terms of  a series of special alpha (i.e., alpha(\omega_k)). What are these alpha(\omega_k)? how do we know this sequence is good enough? That is, how do we know these alpha covers enough possible alphas, and how does alpha(\omega_k) varies with \omega_k?

Line 239:  (alpha(\omega_k)) is such that the equality holds. How do we know such an alpha exist? How do we compute it?

I have some doubts on how to implement the proposed methods. Specifically: Line 234: when computing x_R, \xi\in\Xi(\infty). So how exactly should we compute x_R? What is \mathcal{U} in Line 253 for the linear problem with ellipsoidal uncertainty set?

---

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: **Questions:**
- Line 239: in Theorem 1, $\alpha(\omega_k)$ seems to depend on $x$. How does this work?
- Line 247: what are the two passes exactly?
- Line 299: in Proposition 3, again, the existence of such $\alpha(\omega_k)$ needs to be explained. Is the equality with respect to a specific choice of $x$?

**Major Suggestions:**
- Line 231: define $e$ from the simplex domain.
- Line 329: in Corollary 1, $m$ cannot be arbitrarily large, which makes the probability upper-bounded, so this is not exactly a high probability bound.

**Minor Suggestions:**
- Line 34: need cites in this first paragraph.
- Line 46: also need more cites in the second paragraph.
- Line 91: $E$ is not empty but only includes the zero vector.
- Line 173: explain the connection of this central path to the robust optimization problem earlier on.
soundness: 2
strengths: **Originality:** 
- Their undertaking of efficient identification of Pareto front seems new.  
- The work is a novel combination of well-known techniques.

**Quality:**
- The methods used seem appropriate.
- This is a complete piece of work, with possible future improvements.
- The authors are careful and honest about evaluating both the strengths and weaknesses of their work.

**Clarity:**
- The submission is clearly written.
- It is well organized.

**Significance:**
- The result seems important from a computational efficiency perspective.
- Others (researchers or practitioners) are likely to use the ideas or build on them.
- It provides unique conclusions about existing approaches.
summary: This paper studies approximating the efficient-robust Pareto solutions through proximal point method. It contributes faster computation requirements compared to the literature.
weaknesses: **Originality:** 
- It is not exactly clear how this work differs from previous contributions or if the related work is adequately cited, since there is not substantial comparison.

**Quality:**
- A key point is that the submission has certain issues with technically sound, please see Questions. Similarly, some claims need more support.

**Clarity:**
- It occasionally fails to adequately inform the reader.

**Significance:**
- The difficultness of the task the submission addresses is hard to gauge, considering the rather simple approaches. The comparison with previous work is a bit lacking.
- Thus, it is also not easy to conclude if it advances the state of the art in a demonstrable way.

---

confidence: 2
contribution: 2
initial_rating: 5
presentation: 3
questions: Q1 - Although the computational efficiency claim somewhat makes sense. I am not sure how this is equivalent to generating multiple robust solutions and comparing their results. While your proximal point method generates pareto efficient intermediate solutions, how many of them does it generate? Why is the new computational complexity $2\times T$ and does not include $N$ at all. 

Q2 - How comprehensive is the robust solutions generated by your proximal point based procedure?

Q3 - How to read figure 1 and 2, why are not one-to-one? Are these supposed to be the trajectories followed with the gradient steps?
soundness: 3
strengths: S1 - The paper is generally well written and well presented.

S2 - The technical claims in the paper are generally well explained.

S3 - Experiments are a plus.

S4 - The idea is interesting
summary: This paper presents a proximal point method based procedure to approximate many Pareto efficient robust solutions. This procedure reduces the computational requirement  by a multiplicity of the number of robust solutions to be generated. They go on to show that this procedure can produce exact Pareto efficient robust solutions for a class of optimization problems.
weaknesses: W1 - The literature review is a bit lacking. The comparisons to previous art are inadequate.

W2 - The main novelty is ambiguous. 

W3 - The method seems to be limited in its generation of robust solutions where the radius of the uncertainty sets are not freely selectable.

---

confidence: 4
contribution: 1
initial_rating: 5
presentation: 3
questions: The paper would benefit from numerical experiments comparing the running time of the proposed method against the brute-force approach. The abstract should explicitly state that mean-variance optimization is the paper's primary focus. Additionally, it would be helpful to acknowledge the existence of a closed-form solution to the Markowitz optimization problem. Regarding Figure 1, the substantial discrepancy between the out-of-sample performance of robust Markowitz++ portfolios, despite their similar in-sample performance, requires further explanation.

Typos: 
Line 157 bracket is misplaced
Line 180 by -> be
Figure 1: Porfolio -> Portfolio
soundness: 3
strengths: The paper raises the interesting question of computing the efficient frontier that balances robustness and efficiency, and proposes a novel approach to finding this frontier. The paper provides an original proof that with high probability, the performances of the Pareto efficient robust solutions are bounded by the performances of two proximal point trajectories. The paper demonstrates clear structural organization. The visualizations are informative, and both the core ideas and theorems are presented with clarity.
summary: The paper proves that for robust LPs with uncertain objective functions under the simplex decision domain and ellipsoidal uncertainty sets, the proximal point trajectory are exactly Pareto efficient robust solutions. For robust LPs with a random polyhedron domain, the paper proves that with high probability, the performances of the Pareto efficient robust solutions are bounded by the performances of two proximal point trajectories. Numerical experiments on portfolio optimization and adversarially robust deep learning are provided.
weaknesses: The paper claims that the computational cost is reduced from N*T to 2*T. However, in each iteration of the proposed proximal point method for portfolio optimization, a quadratic optimization problem needs to be solved. Thus, the total computational cost is not reduced compared to the brute-force method of solving the problem for multiple alphas.

Meta-review: A proximal point method based method is proposed to generate a series of Pareto efficient solutions with computational efficiency. However, the analysis is limited to a particular form of loss functions and is not obvious to the reviewers how to generalize to more general cases. Numerical results do not seem to be very convincing either.

Decision: Reject