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theses

We investigate the non-negative univariate polynomial conic optimization (uPCO) problem from the perspective of applications and the algorithms. We start by considering the applications of uPCO, specifically in: 1- Time-variant network flow problems, and 2- Non-parametric estimation under shape constraints with splines. Regarding algorithms, we use non-symmetric interior point methods (IPMs) to solve this conic optimization problem. It is well known that uPCO can be formulated as a semidefinite programming (SDP) problem, and therefore, it can be solved by available software for SDP. However, doing so will result in squaring the number of decision variables, and thus it is impractical even for moderate size problems. In addition, straightforward SDP formulation involves numerically unstable processes. Regarding the latter issue, we propose an orthogonal change of basis. Using the Chebyshev polynomials (which form an orthogonal basis), uPCO problems with significantly higher dimensions can be solved. As for the former issue, we propose two direct non-symmetric interior-point algorithms, by specializing the non-symmetric homogeneous self-dual predictor-corrector (HSD P-C) IPM (proposed by Skajaa-Ye 2015) and a Mehrotra version (i.e. HSD M-P-C IPM) of this algorithm (proposed by Akle-Ye 2015). We consider implementing these algorithms in two approaches. In the first approach, we develop these IPMs for the dual of the uPCO problem, i.e., the univariate moment conic optimization (uMCO) problem, where the algorithms can be utilized by the efficient barrier function of the moment cone. In the second approach, we consider developing the previous algorithms directly for the uPCO problem, by utilizing the algorithms by the Faybusovich universal barrier function of the non-negative univariate polynomial cone. We present numerical results of our implementations of these algorithms for each approach and a comparison among them. Next, we consider a general conic optimization problem which contains the non-negative polynomial conic constraints, as well as second order and linear conic constraints. We propose a unified non-symmetric HSD IPM for this problem. Finally, we present that the numerical results of our implementations are comparable to the results of the symmetric HSD IPM for the symmetric formulation of the general problem, without the need to square the non-negative polynomial variables.

ETD_8951

Ph.D.

Includes bibliographical references

by Mohammad Mehdi Ranjbar

doi:10.7282/T3Z32335

ETD doctoral

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