We propose a homogeneous primal-dual interior-point method to solve sum-ofsquares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables when the degree of sum-of-squares polynomials is large. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that the proposed method is several orders of magnitude faster than the SDP-based approach for optimization problems over high-degree sum-of-squares polynomials.

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.

For an integer linear program, Gomory’s corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation. In this paper, we do not relax these nonnegativity constraints. We generalize a classical result of Gomory and Johnson characterizing minimal cut-generating functions in terms of subadditivity, symmetry, and periodicity. Our result is based on the notion of generalized symmetry condition. We also prove a 2-slope theorem for extreme cut-generating functions in our setting, in the spirit of the 2-slope theorem of Gomory and Johnson.

In this paper we study general two-term disjunctions on affine cross-sections of the secondorder cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids. Our approach relies on the work of Kılınç-Karzan and Yıldız which considers general two-term disjunctions on the second-order cone.

Background: Periodontal regeneration is dependent on the uninterrupted adhesion, maturation and absorption of fibrin clots to a periodontally compromised root surface. The modification of the root surface with different agents has been used for better fibrin clot formation and blood cell attachment. It is known that Er:YAG laser application on dentin removes the smear layer succesfully.Aim: The aim of this study is to observe blood cell attachment and fibrin network formation following ER:YAG laser irradiation on periodontally compromised root surfaces in comparison to chemical root conditioning techniques in vitro.Materials and methods: 40 dentin blocks prepared from freshly extracted periodontally compromised hopeless teeth. Specimens were divided in 5 groups; those applied with PBS, EDTA, Citric acid and Er:YAG. They were further divided into two groups: those which had received these applications, and the control group. The specimens were evaluated with scanning electron microscope and micrographs were taken. Smear layer and blood cell attachment scoring was performed.Results: In the Er:YAG laser applied group, smear layer were totally removed. In the blood applied specimens, better fibrin clot formation and blood cell attachment were observed in the Er:YAG group. In the group that had been applied with citric acid, the smear layer was also removed. The smear layer could not be fully removed in the EDTA group.Conclusion: Er:YAG laser application on the root dentin seems to form a suitable surface for fibrin clot formation and blood cell attachment. Further clinical studies to support these results are necessitated.

This note settles an open problem about cut-generating functions, a concept that has its origin in the work of Gomory and Johnson from the 1970's and has received renewed attention in recent years.

Abstract. Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, can potentially lower treatment side effects without compromising tumor control. This can be achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the surrounding tissue. Plan optimization for such treatments is based on biologically effective dose (BED); however, this leads to computationally challenging nonconvex optimization problems. Optimization methods that are in current use yield only locally optimal solutions, and it has hitherto been unclear whether these plans are close to the global optimum. We present an optimization framework to compute rigorous bounds on the maximum achievable normal tissue BED reduction for spatiotemporal plans.The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising any other treatment objective. The BED-based treatment plan optimization problems are formulated as quadratically constrained quadratic programming (QCQP) problems.First, a conventional, uniformly fractionated reference plan is computed using convex optimization. Then, a second, nonconvex, QCQP model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED, subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form arXiv:1712.03046v1 [physics.med-ph] 8 Dec 2017Spatiotemporal fractionation with bounds on the achievable benefit 2 of a semidefinite programming (SDP) problem, which provides a rigorous lower bound on the lowest achievable mean liver BED.The method is presented on 5 cases with distinct geometries. The computed spatiotemporal plans achieve 12-35 percent mean liver BED reduction over the optimal uniformly fractionated plans. This reduction corresponds to 79-97 percent of the gap between the mean liver BED of the uniform reference plans and our lower bounds on the lowest achievable mean liver BED. The results indicate that spatiotemporal treatments can achieve substantial reductions in normal tissue dose and BED, and that local optimization techniques provide high-quality plans that are close to realizing the maximum potential normal tissue dose reduction.