Here are my papers. The titles are links to the content (if available).

**Under Review**

*Distributed Projections onto a Simplex*

Submitted to Mathematical Programming Computations.

Y. Dai, C. Chen

Projecting a vector onto a simplex is a well-studied problem that arises in a wide range of optimization problems. Numerous algorithms have been proposed for determining the projection; however, all but one of these algorithms are serial. We address this gap by developing a method that preprocesses the input vector by decomposing and distributing it across multiple processors for local projection. Our method is especially effective when the projection is highly sparse; which is the case, for instance, in large-scale problems with i.i.d. entries. Moreover, the method can be adapted to work with a broad range of serial algorithms from the literature. We fill in theoretical gaps in serial algorithm analysis, and develop similar results for our parallel analogues. Numerical experiments conducted on a wide range of large-scale instances demonstrate the practical effectiveness of the method.

Nonnegative Tensor Completion via Integer Optimization

Submitted to The Thirty-ninth International Conference on Machine Learning

C. Bugg, C. Chen, A. Aswani

Unlike matrix completion, no algorithm for the tensor completion problem has so far been shown to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative tensors. We prove that our algorithm converges in a linear (in numerical tolerance) number of oracle steps, while achieving the information-theoretic rate. Our approach is to define a new norm for nonnegative tensors using the gauge of a specific 0-1 polytope that we construct. Because the norm is defined using a 0-1 polytope, this means we can use integer linear programming to solve linear separation problems over the polytope. We combine this insight with a variant of the Frank-Wolfe algorithm to construct our numerical algorithm, and we demonstrate its effectiveness and scalability through experiments.

**Published**

*Outer-Product-Free Sets for Polynomial Optimization and Oracle-Based Cuts*

Mathematical Programming, 2020.

D. Bienstock, C. Chen, G. Muñoz

Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S∩P, where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S we construct a convergent pure cutting plane algorithm; if the initial relaxation is a polytope, the algorithm is shown to converge. Cuts are generated from convex forbidden zones, or S-free sets derived from the oracle. We also consider the special case of polynomial optimization. Polynomial optimization may be represented using a symmetric matrix of variables, and in this lifted representation we can let S be the set of matrices that are real, symmetric outer products. Accordingly we develop a theory of outer-product-free sets. All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify two families of such sets. These families can be used to generate intersection cuts that can separate any infeasible extreme point of a linear programming relaxation in polynomial time. Moreover, the oracle-based approach can be strengthened and implemented in polynomial-time for polynomial optimization.

*Intersection Cuts for Polynomial Optimization*

IPCO, 2019

D. Bienstock, C. Chen, G. Muñoz

We consider dynamically generating linear constraints (cutting planes) to tighten relaxations for polynomial optimization problems. Many optimization problems have feasible set of the form S∩P, where *S* is a closed set and *P* is a polyhedron. Integer programs are in this class and one can construct intersection cuts using convex “forbidden” regions, or *S*-free sets. Here, we observe that polynomial optimization problems can also be represented as a problem with linear objective function over such a feasible set, where *S* is the set of real, symmetric matrices representable as outer-products of the form xx^T. Accordingly, we study outer-product-free sets and develop a thorough characterization of several (inclusion-wise) maximal intersection cut families. In addition, we present a cutting plane approach that guarantees polynomial-time separation of an extreme point in P∖S using our outer-product-free sets. Computational experiments demonstrate the promise of our approach from the point of view of strength and speed.

*A Spatial Branch-and-Cut Algorithm for Nonconvex QCQP with Bounded Complex Variables
*Mathematical Programming, 2016

C. Chen, A. Atamtürk and S.S. Oren

We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of CQCQP. These valid inequalities are derived from the convex hull description of a nonconvex set of 2×2 positive semidefinite Hermitian matrices subject to rank-one constraint. We propose branching rules based on an alternative to a rank-one constraint that allows for local measurement of constraint violation. Closed-form bound tightening procedures are used to reduce the domain of the problem. We apply the algorithm to solve the Alternating Current Optimal Power Flow problem with complex variables and the Box-constrained Quadratic Programming problem with real variables.

*Bound Tightening for the Alternating Current Optimal Power Flow Problem*

IEEE Transactions on Power Systems, 2015

C. Chen, A. Atamtürk and S.S. Oren

We consider the Alternating Current Optimal Power Flow (ACOPF) problem, formulated as a nonconvex Quadratically-Constrained Quadratic Program (QCQP) with complex variables. ACOPF may be solved to global optimality with a semidefinite programming (SDP) relaxation in cases where its QCQP formulation attains zero duality gap. However, when there is positive duality gap, no optimal solution to the SDP relaxation is feasible for ACOPF. One way to find a global optimum is to partition the problem using a spatial branch- and-bound method. Tightening upper and lower variable bounds can improve solution times in spatial branching by potentially reducing the number of partitions needed. We propose special- purpose closed-form bound tightening methods to tighten limits on nodal powers, line flows, phase angle differences, and voltage magnitudes. Computational experiments are conducted using a spatial branch-and-cut solver. We construct variants of IEEE test cases with high duality gaps to demonstrate the effectiveness of the bound tightening procedures.

*Robust Portfolio Selection for Index Tracking
*Computers & Operations Research, 2012

C. Chen and R.H. Kwon

We develop a robust portfolio selection model for tracking a market index using a subset of its assets. The model is a 0–1 integer program that seeks to maximize similarity between selected assets and the assets of the target index. We allow uncertainty in the objective function by using a computationally tractable robust framework that can control the conservativeness of the solution. This protects against worst-case realizations of potential estimation errors and other deviations. Out-of-sample experiments using the S&P 100 demonstrate the advantages of the robust model. Compared to portfolios constructed with the nominal model, moderately conservative robust portfolios are shown to have lower tracking error and risk profiles that are more similar to the target index.

**Dissertation Abstract**

Title:* Branch-and-Cut for Nonlinear Power Systems Problems
*Dissertation Committee: S.S. Oren (Chair), A. Atamtürk, K. Poolla, D. Tse

This dissertation is concerned with the design of branch-and-cut algorithms for a variety of nonconvex nonlinear problems pertaining to power systems operations and planning. By understanding the structure of specific problems, we can leverage powerful commercial optimization solvers designed for convex optimization and mixed-integer programs. The bulk of the work concerns the Alternating Current Optimal Power Flow (ACOPF) problem. The ACOPF problem is to find a minimum cost generation dispatch that will yield flows that can satisfy demand as well as various engineering constraints. A standard formulation can be posed as a nonconvex Quadratically Constrained Quadratic Program with complex variables. We develop a novel spatial branch-and-bound algorithm for generic nonconvex QCQP with bounded complex variables that relies on a semidefinite programming (SDP) relaxation strengthened with linear valid inequalities. ACOPF-specific domain reduction or bound tightening techniques are also introduced to improve the algorithm’s convergence rate. We also introduce second-order conic valid inequalities so that any SDP can be outer-approximated with conic quadratic cuts and test the technique on ACOPF. Another application is the incorporation of convex quadratic costs in unit commitment, which is a multi-period electric generation scheduling problem. We show that conic reformulation can both theoretically and practically improve performance on this mixed-integer nonlinear problem. We conclude with methods for approximating a mixed-integer convex exponential constraint. Applications include capital budgeting, the system reliability redundancy problem, and feature subset selection for logistic regression.