Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

Abstract

Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.

Cardinality-constrained Distributionally Robust Portfolio Optimization

“…We extend the cutting-plane algorithm [7] with the aim of solving the upper-level problem (8). Let θ LB be a lower bound of the optimal objective value of Problem (8); this bound can easily be calculated by solving a continuous relaxation version of Problem (7). Our cutting-plane algorithm starts with the initial feasible region:…”

“…( 11). If f (z t ) ≤ θ t + ε holds with sufficiently small ε ≥ 0, then z t is an ε-optimal solution to Problem (8), which means that…”

“…where f ⋆ is the optimal objective value of Problem (8). In this case, we terminate the algorithm with the ε-optimal solution z t .…”

“…Algorithm 1 Cutting-plane algorithm for solving Problem (8) Step 0 (Initialization) Let ε ≥ 0 be a tolerance for optimality. Define the feasible region F 1 as in Eq.…”

“…Kobayashi et al [35] developed a bilevel cutting-plane algorithm to solve cardinality-constrained mean-CVaR portfolio optimization problems. The cutting-plane algorithm [9] was also applied to sparse principal component analysis [8] and sparse inverse covariance estimation [13].…”