insight - Optimization algorithms

### Accelerated Optimization of the Linear-Quadratic Regulator Problem

This paper introduces an accelerated optimization framework for solving the linear-quadratic regulator (LQR) problem, which is a landmark problem in optimal control. The authors present novel continuous-time and discrete-time algorithms that achieve Nesterov-optimal convergence rates for the state-feedback LQR (SLQR) problem. For the output-feedback LQR (OLQR) problem, a Hessian-free accelerated framework is proposed that can find an ϵ-stationary point with second-order guarantee in a time of O(ϵ^(-7/4) log(1/ϵ)).

### Efficient Approximation Algorithm for Multi-Dimensional Scaling via Sherali-Adams LP Hierarchy

We present the first quasi-polynomial time approximation algorithm for the Kamada-Kawai formulation of Multi-Dimensional Scaling, achieving a solution with cost O(log Δ) · OPT^Ω(1) + ε in time nO(1) · 2^poly((log(Δ)/ε)), where Δ is the aspect ratio of the input dissimilarities.

### Algebraic Proofs of Path Disconnectedness using Time-Dependent Barrier Functions

The core message of this article is that path disconnectedness between two sets X0 and X1 within a larger set X can be certified through the existence of a time-dependent barrier function that separates the sets. This barrier function is a necessary and sufficient condition for path disconnectedness under compactness assumptions.

### Linear Convergence of Forward-Backward Accelerated Algorithms for Strongly Convex Functions Without Knowledge of the Modulus of Strong Convexity

The paper establishes the linear convergence of NAG and FISTA for strongly convex functions without requiring any prior knowledge of the modulus of strong convexity.

### Efficient Combinatorial Algorithm for Computing Explicit Solutions to Multi-Parametric Quadratic Programs

The paper proposes a combinatorial method for efficiently computing explicit solutions to multi-parametric quadratic programs, which can be used to compute explicit control laws for linear model predictive control. The method is based on exploring a connected graph of combinatorially adjacent active sets, avoiding the need for demanding geometrical operations.

### Efficient Difference of Submodular Minimization via DC Programming Algorithms

Difference of submodular (DS) minimization can be equivalently formulated as the minimization of the difference of two convex (DC) functions. The authors introduce variants of the DC algorithm (DCA) and its complete form (CDCA) to efficiently solve the DC program corresponding to DS minimization, and establish new connections between the two problems to obtain stronger theoretical guarantees.

### Efficient Algorithms for Dynamic Assortment Optimization under Multinomial Logit Choice Model

We develop a unified algorithmic framework that provides provable approximation guarantees for dynamic assortment optimization problems under the Multinomial Logit choice model, improving upon the state-of-the-art results. Our algorithms address both the dynamic assortment problem without personalization and the dynamic assortment problem with personalization, and can handle uncertainty in the total number of customers.

### Efficient Optimization of Strongly Convex Functions with Linear Constraints using Accelerated Randomized Bregman-Kaczmarz Method

The authors propose an accelerated randomized Bregman-Kaczmarz method to efficiently solve linearly constrained optimization problems with strongly convex (possibly non-smooth) objective functions. They provide theoretical analysis showing linear convergence rates and demonstrate the superior efficiency of the proposed method compared to existing approaches.

### Efficient Zeroth-Order Bilevel Optimization via Gaussian Smoothing

This paper proposes a fully zeroth-order stochastic approximation method for solving bilevel optimization problems, where neither the upper/lower objective values nor their unbiased gradient estimates are available. The authors use Gaussian smoothing to estimate the first- and second-order partial derivatives of the functions with two independent block of variables, and establish non-asymptotic convergence analysis and sample complexity bounds for the proposed algorithm.

### Contention Resolution Schemes for Hypergraph Matching, Knapsack, and k-Column Sparse Packing Problems

The contention resolution framework is a versatile rounding technique used to solve constrained submodular function maximization problems. This work applies this framework to the hypergraph matching, knapsack, and k-column sparse packing problems, providing non-constructive lower bounds on the correlation gap and constructing monotone contention resolution schemes.