Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
In this dissertation, novel approaches for solving convex nonsmooth optimization, variational inequalities and inclusion problems are studied. The main contributions of the dissertation are given in Chapter 4 and Chapter 5. The two proposed iterations in Chapter 4, Half-Extragradient algorithm (HEG) and its accelerated version, are a natural modiﬁcation of the classical Extragradient algorithm (EG)
when the composite objective function is a sum of three convex functions. EG evaluates the smooth operator twice per iteration via proximal mappings, and also, it allows larger step sizes. One of the main advantages of the proposed scheme is to avoid evaluating an
extragradient step per iteration. The convergence, sublinearity and complexity analyses of the generated sequences are established by forcing Fej´er monotonicity. We provide a sublinear rate of O(1/k) for the HEG and extend it to O(1/k2) for its associate accelerated version. Moreover, still in this chapter, the classical extragradient algorithm for solving variational inequalities for the sum of two monotone operators is modiﬁed. The proposed method only evaluates the cocoercive operator once per iteration, and its convergence results are provided with one operator being Lipschitz continuous. In Chapter 5, a conceptual algorithm is proposed to modifying the popular Tseng’s forward-backward-forward (FBF) splitting method for solving monotone inclusions. It is well-known that the FBF improves the convergence properties of the classical forward backward (FB) splitting iteration by adding an extra forward step. The proposed conceptual algorithm generalizes the forward-backward-half-forward (FBHF) iteration (which recovers the FBF) by introducing two diﬀerent projection (forward) steps. Both proposed variants work eﬃciently with relaxing the Lipschitz continuity of the smooth operator and the cocoercivity of the other operator. Convergence analysis of both variants is presented, the ﬁrst variant is a generalization of the FBHF iteration. In the second variant, the generated sequence is entirely contained in a ball with diameter equal to the distance between the initial state and the solution set, and converges strongly to the optimal solution. Reminding that
only weak convergence is known for the FBHF splitting method.
Hazaimah, Oday, "Projective Splitting Methods For Maximal Monotone Mappings in Hilbert Spaces" (2020). Graduate Research Theses & Dissertations. 7102.
Northern Illinois University
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