Convex Optimization, Domain-Driven Form

Convex optimization is an important special case of mathematical optimization applications in a wide range of disciplines. Convex optimization's powerful and elegant theory has been coupled with faster and more reliable numerical linear algebra software and powerful computers to spread its applications over many fields such as (1) data science,(2) control theory and signal processing, (3) relaxation and randomization for hard nonconvex problems, and (4) robust optimization.

infea-start We say a problem is in the Domain-Driven form if the underlying convex set is given as the domain of a convex function which is a self-concordant (s.c.) barrier. We have designed and analyzed infeasible-start primal-dual interior-point algorithms for problems given in the Domain-Driven form. Infeasible-start means our algorithms can return the status of the given problem with robust certificates of approximate optimality, unboundedness, and infeasibility. These algorithms are the foundation of a software packge Domain-Driven Solver (DDS). Even though we have meticulously implemented the Hessian-vector product for every constraint type, there are still many possibilities to explore. To improve the performance of DDS and invent new optimization techniques, I am also working to leverage second-order information.

Domain-Driven Solver (DDS)

We have released DDS 2.1 in July 2022. This version is accepted for publication in Mathematical Programming Computation. DDS accepts many interesting classes of function/set constraints. You can solve several classes of convex optimization problems using DDS 2.1, ranging from LP and SDP to quantum entorpy and hyperbolic programming. DDS is also easily expandable, and by the discovery of a s.c. barrier, a new set constraint can be added to DDS.

Operations of Modern Power Networks

In the context of the rapidly transforming power networks, characterized by millions of controllable nodes, achieving effective solutions necessitates a comprehensive grasp of the system's complexities and advanced mathematical principles. Leveraging a fusion of optimization and engineering expertise, I aim to pioneer innovative strategies, adapting new scalable optimization techniques and data-driven approaches for modern power networks. The primary goal is designing robust algorithms to heighten efficiency, reliability, and adaptability in the face of evolving, decarbonized energy paradigms.

Optimization of quantum entropy and quantum relative entropy

qu-inf Quantum entropy (QE) and quantum relative entropy (QRE) have applications in quantum information processing. DDS accepts constraints involving quantum entropy and quantum relative entropy and significantly outperforms methods that approximate these functions by SDP constraints. A new version DDS 2.2 will be released soon which has been significantly improved for solving QRE problems. Despite acceptable performance, there is still room in DDS for performance improvement. The barrier function we use in DDS for quantum relative entropy is not a proven s.c. barrier. We are currently working to prove a conjectured s.c. barrier for the epigraph of quantum relative entropy function. In the application part, there are many open questions mostly related to efficiently evaluating the derivatives of the barrier (matrix) functions used for quantum entropy and relative entropy.

Hyperbolic programming

qu-inf Hyperbolic programming has recently attracted attention for its connection to the sum-of-squares methods and generalized Lax conjecture. Accepting hyperbolic constraints is a major contribution in DDS. DDS accepts polynomials in three different formats: list of monomials, straight-line program, and determinantal representation. To scale DDS with the size of the input hyperbolic polynomials in each format, some interesting open questions have to be answered. For example, for the straight-line program, we are working to implement an automatic differentiation scheme for evaluating the derivatives of the polynomials. Automatic differentiation has been studied extensively for its applications in machine learning.

Sum-of-Squares and SDP Complexity

In a very interesting theoretical project, we are using the theory of sum-of-squares of polynomials to attack the Turing machine complexity of semidefinite programs (SDP). Even though an approximate solution for an SDP can be found in polynomial time and having an SDP formulation is desirable in many applications, very little is known about the complexity of SDP in the Turing machine model. Even the question of whether SDP feasibility is in the NP class or not is open. In the first part of this project, we have been studying the complexity of SDP problems arise form the univariate nonnegative polynomials and trying to deepen our understanding of their structure.

Nonnegative polynomials and SOS problems have been studied for more than a century; however, numerous new applications in optimization and machine learning have made them very popular among optimizers. The SOS method has been used to improve guarantees for many approximations algorithms and become a new tool in computational complexity. To mention new applications in machine learning, this method has given new bounds for sparse vector recovery and dictionary learning.