Approximating Pareto Frontiers in Stochastic Multi-Objective Optimization via Hashing and Randomization

Abstract

Stochastic Multi-Objective Optimization (SMOO) is critical for decision-making trading off multiple potentially conflicting objectives in uncertain environments. SMOO aims at identifying the Pareto frontier, which contains all mutually non-dominating decisions. The problem is highly intractable due to the embedded probabilistic inference, such as computing the marginal, posterior probabilities, or expectations. Existing methods, such as scalarization, sample average approximation, and evolutionary algorithms, either offer arbitrarily loose approximations or may incur prohibitive computational costs. We propose XOR-SMOO, a novel algorithm that with probability $1-δ$, obtains $γ$-approximate Pareto frontiers ($γ>1$) for SMOO by querying an SAT oracle poly-log times in $γ$ and $δ$. A $γ$-approximate Pareto frontier is only below the true frontier by a fixed, multiplicative factor $γ$. Thus, XOR-SMOO solves highly intractable SMOO problems (\#P-hard) with only queries to SAT oracles while obtaining tight, constant factor approximation guarantees. Experiments on real-world road network strengthening and supply chain design problems demonstrate that XOR-SMOO outperforms several baselines in identifying Pareto frontiers that have higher objective values, better coverage of the optimal solutions, and the solutions found are more evenly distributed. Overall, XOR-SMOO significantly enhanced the practicality and reliability of SMOO solvers.

Figures (8)

• Figure 1: Solving SMOO problems via querying satisfiability oracles. (Left) In multi-objective optimization (MOO), Papadimitriou et al. papadimitriou2000approximability lays down a multiplicative grid, where adjacent grid points are separated by $\gamma$. Then, for every grid point, a SAT oracle is queried to determine if a solution exists such that each of its objective value exceeds the grid point’s value at the corresponding dimension. SAT oracles' responses split the entire region into a (SAT) and a (UNSAT) region. The top of all green points form a $\gamma$-approximate Pareto frontier. (Right) In SMOO, with high probability, our developed probabilistic oracle makes the correct SAT/UNSAT decisions only when the objective values exceed (or lack behind) the queried threshold by a fixed multiplicative constant. This brings in a third, intermediate uncertain region (shown in blue). However, because its width can be controlled, the top of all green points still form a $\gamma$-approximate Pareto frontier.
• Figure 2: Illustration of our approach for solving SMOO problems in a two-objective maximization setting. Task: Both objectives involve model counting, are defined over decision variables $x$ and latent variables $y_1, y_2$. The orange curve shows the Pareto frontier. Step 1: By discretizing objective space multiplicatively by a factor of 2, optimization is converted into a set of SAT queries asking whether the thresholds at each grid point are jointly achievable, separating points into SAT (green) and UNSAT (red) regions. The true Pareto frontier is sandwiched between the adjacent green/red points. Step 2: Since each SAT query is intractable, we instead use a probabilistic SAT oracle providing two guarantees: (1) correctness of the SAT/UNSAT outcome; and (2) if SAT, the returned solution achieves tightly guaranteed objective values. These guarantees yield a sketch of the Pareto frontier via the SAT/UNSAT boundary, and the corresponding solutions $x$ collectively form an approximate Pareto frontier.
• Figure 3: Example of solving a two-objective optimization problem via discretized decision problems, assuming exact inference were possible. The SAT boundary solutions would form a $2$-approximate Pareto frontier.
• Figure 4: Converting a weighted function $f(\mathbf{x}, \mathbf{y})$ into unweighted model counting $\sum_{\mathbf{z}} \hat{f}(\mathbf{x}, \mathbf{y}, \mathbf{z}; b)$. (Left) The function $f$ is normalized and uniformly discretized into $2^b$ integer levels, where $b \in \mathbb{Z}_{>0}$ is a user-specified discretization precision. The red bar shows an example $f(\mathbf{x}_0, \mathbf{y}_0)$, which is discretized to the binary integer $(1,0,0,0)_2$ (or $8$ in decimal). (Right) We construct an unweighted function $\hat{f}$ such that, when $\mathbf{z} \in \{0,1\}^b$ is interpreted as a binary integer, $\hat{f}(\mathbf{x}, \mathbf{y}, \mathbf{z}; b) = 1$ if $\mathbf{z}$ is no greater than the discretized $f(\mathbf{x}, \mathbf{y})$, and $0$ otherwise. Then $f(\mathbf{x}, \mathbf{y})$ can be directly computed as $\sum_{\mathbf{z}} \hat{f}$, e.g., the red bar shows that $\sum_{\mathbf{z}} \hat{f}(\mathbf{x}_0, \mathbf{y}_0, \mathbf{z}; b) = (1,0,0,0)_2$ exactly.
• Figure 5: Road Network Strengthening to Mitigate Seasonal Disruptions. Green edges denote operational road segments, while red edges represent disrupted segments under seasonal events. Yellow and blue regions indicate high-impact disruption areas in summer (e.g., heavy rain, extreme heat) and winter (e.g., heavy snow, extreme cold), respectively. The planner selects a limited set of road segments to strengthen in order to maintain connectivity between the source node $S$ and the target node $T$ under seasonal disruptions.

Theorems & Definitions (25)

• Definition 1: Pareto Frontier
• Definition 2: $\gamma$-Approximate Pareto Frontier papadimitriou2000approximability
• Lemma 1: XOR Counting jerrum1986randomGomes06XORCountingErmon13Wish
• Theorem 2: Papadimitriou and Yannakakis papadimitriou2000approximability
• Lemma 3
• proof
• Definition 3: Probabilistic SAT Oracle
• Theorem 4: High-quality Pareto frontier curve
• proof
• Theorem 5: Approximate Pareto frontier