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1. A framework for quadratic form maximization over convex sets through nonconvex relaxations

   https://doi.org/10.1145/3406325.3451128  

   Bhattiprolu, Vijay ; Lee, Euiwoong ; Naor, Assaf ( January 2021 , STOC'21)

   null (Ed.)

   We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an origin-symmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n. However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows us to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex when K has a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators. Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization. 

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2. Robust Optimization-based Motion Planning for high-DOF Robots under Sensing Uncertainty

   https://doi.org/10.1109/ICRA48506.2021.9560917  

   Quintero-Pena, Carlos ; Kyrillidis, Anastasios ; Kavraki, Lydia E. ( May 2021 , 2021 IEEE International Conference on Robotics and Automation (ICRA))

   null (Ed.)

   Motion planning for high degree-of-freedom (DOF) robots is challenging, especially when acting in complex environments under sensing uncertainty. While there is significant work on how to plan under state uncertainty for low-DOF robots, existing methods cannot be easily translated into the high-DOF case, due to the complex geometry of the robot’s body and its environment. In this paper, we present a method that enhances optimization-based motion planners to produce robust trajectories for high-DOF robots for convex obstacles. Our approach introduces robustness into planners that are based on sequential convex programming: We reformulate each convex subproblem as a robust optimization problem that “protects” the solution against deviations due to sensing uncertainty. The parameters of the robust problem are estimated by sampling from the distribution of noisy obstacles, and performing a first-order approximation of the signed distance function. The original merit function is updated to account for the new costs of the robust formulation at every step. The effectiveness of our approach is demonstrated on two simulated experiments that involve a full body square robot, that moves in randomly generated scenes, and a 7-DOF Fetch robot, performing tabletop operations. The results show nearly zero probability of collision for a reasonable range of the noise parameters for Gaussian and Uniform uncertainty. 

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3. Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

   https://doi.org/10.1287/moor.2020.1100  

   Gutekunst, Samuel C. ; Williamson, David P. ( July 2021 , Mathematics of Operations Research)

   null (Ed.)

   The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP. 

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4. A Computational Framework for Solving Nonlinear Binary Optimization Problems in Robust Causal Inference

   https://doi.org/10.1287/ijoc.2022.1226  

   Islam, Md Saiful ; Morshed, Md Sarowar ; Noor-E-Alam, Md. ( November 2022 , INFORMS Journal on Computing)

   Identifying cause-effect relations among variables is a key step in the decision-making process. Whereas causal inference requires randomized experiments, researchers and policy makers are increasingly using observational studies to test causal hypotheses due to the wide availability of data and the infeasibility of experiments. The matching method is the most used technique to make causal inference from observational data. However, the pair assignment process in one-to-one matching creates uncertainty in the inference because of different choices made by the experimenter. Recently, discrete optimization models have been proposed to tackle such uncertainty; however, they produce 0-1 nonlinear problems and lack scalability. In this work, we investigate this emerging data science problem and develop a unique computational framework to solve the robust causal inference test instances from observational data with continuous outcomes. In the proposed framework, we first reformulate the nonlinear binary optimization problems as feasibility problems. By leveraging the structure of the feasibility formulation, we develop greedy schemes that are efficient in solving robust test problems. In many cases, the proposed algorithms achieve a globally optimal solution. We perform experiments on real-world data sets to demonstrate the effectiveness of the proposed algorithms and compare our results with the state-of-the-art solver. Our experiments show that the proposed algorithms significantly outperform the exact method in terms of computation time while achieving the same conclusion for causal tests. Both numerical experiments and complexity analysis demonstrate that the proposed algorithms ensure the scalability required for harnessing the power of big data in the decision-making process. Finally, the proposed framework not only facilitates robust decision making through big-data causal inference, but it can also be utilized in developing efficient algorithms for other nonlinear optimization problems such as quadratic assignment problems. History: Accepted by Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation of the National Science Foundation [Grant 2047094]. Supplemental Material: The online supplements are available at https://doi.org/10.1287/ijoc.2022.1226 . 

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5. Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

   https://doi.org/10.1007/s10107-023-01966-0  

   He, Ziyu ; Han, Shaoning ; Gómez, Andrés ; Cui, Ying ; Pang, Jong-Shi ( May 2023 , Mathematical Programming)

   This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “$$\ell _0$$${\ell }_0$-path” where the discontinuous$$\ell _0$$${\ell }_0$-function provides the exact sparsity count; the “$$\ell _1$$${\ell }_1$-path” where the$$\ell _1$$${\ell }_1$-function provides a convex surrogate of sparsity count; and the “capped$$\ell _1$$${\ell }_1$-path” where the nonconvex nondifferentiable capped$$\ell _1$$${\ell }_1$-function aims to enhance the$$\ell _1$$${\ell }_1$-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped$$\ell _1$$${\ell }_1$-path. Our study of the capped$$\ell _1$$${\ell }_1$-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped$$\ell _1$$${\ell }_1$-regularized problem offers interesting theoretical properties and practical compromise between the$$\ell _0$$${\ell }_0$-path and the$$\ell _1$$${\ell }_1$-path. Indeed, while the$$\ell _0$$${\ell }_0$-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of$$\ell _1$$${\ell }_1$-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped$$\ell _1$$${\ell }_1$-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

    

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