In this talk, we propose a procedure for listing KKT(Karush-Kuhn-Tucker) points of a quadratic reverse convex programming problem (QRC) whose feasible set is expressed as the area excluded the interior of a convex set from another convex set. Several types of iterative solution methods for solving (QRC) have been proposed by many other researchers. However, such algorithms are not effective in the case where the dimension of variables is so large. For such problems, we introduce an algorithm for listing KKT points of (QRC). Moreover, by combining our algorithm into a branch and bound procedure, we obtain most of KKT points of (QRC). By utilizing our algorithm, we can calculate most of locally optimal solutions contained in the intersection of the boundaries of convex sets defining the feasible set. Moreover, by choosing a calculated locally optimal solution having the smallest value of the objective function, we can obtain an approximate solution of a globally optimal solution is obtained. Furthermore, to improve calculation efficiency of our algorithm, we propose an update method of Lagrange multipliers for convex constraint conditions. The effectiveness of the improvement has been shown by the result of the computer experiment.

We discuss continuous formulations for several important discrete optimization problems. The proposed formulations can be used to develop analytical bounds as well as effective algorithms for some of the problems. Moreover, a novel hierarchy of nonconvex continuous reformulations of discrete optimization problems is discussed.

RNA viruses exhibit high mutation rates, leading to the formation of new viral strains through point mutations, insertions, and deletions, collectively known as a viral quasispecies. Despite the advancements enabled by high-throughput sequencing (HTS) in quasispecies analysis, challenges arise from sequencing errors and limited read lengths. The reconstruction of viral quasispecies is particularly challenging due to the unknown number of strains, non-uniform strain frequencies, and sparse observed data with missing elements. This difficulty is further amplified when the diversity among the strains is small. Our research introduces an algorithm that employs a clustering and tensor factorization framework to analyze sparse tensor data derived from aligned sequencing reads, aiming to reconstruct viral quasispecies characterized by highly uneven component frequencies. Specifically, we utilize an iterative clustering framework to estimate the number of strains. Within each iteration, we utilize an alternating least-square method, incorporating missing data imputation, to solve the corresponding sparse tensor factorization problem and reconstruct the strains of quasispecies. Our results, obtained from simulated datasets with small distances between strains, demonstrate the capability of our method to accurately estimate frequencies and reconstruct strains characterized by highly uneven frequencies. Generally, our method outperforms state-of-the-art approaches, particularly at diversities of around 1\%.

When faced with multiple minima of an "inner-level" convex optimization problem, the convex bilevel optimization problem selects an optimal solution which also minimizes an auxiliary "outer-level" convex objective of interest. Bilevel optimization requires a different approach compared to single-level optimization problems since the set of minimizers for the inner-level objective is not given explicitly. Furthermore, Slater constraint qualification is never satisfied for this problem, thus strong duality is not guaranteed to hold in many cases. In this paper, we propose two new projection-free methods for convex bilevel optimization which require only a linear optimization oracle over the base domain. The first method employs the iterative regularization idea to form a blended objective with regularization parameter determined by a fixed schedule. The second method interprets the regularization parameter as a Lagrangian dual variable, and employs estimates of the inner-level optimal objective to allow for dynamic adjustment of this parameter throughout the course of the algorithm. We provide convergence guarantees for both inner- and outer-level objectives that hold under our proposed projection-free methods. In particular, we highlight how our guarantees are affected by the presence or absence of an optimal dual solution. Lastly, we conduct numerical experiments that demonstrate the performance of the proposed methods.