17 research outputs found

    Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods

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    Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the 00 dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but more involved, description of the real radical for systems with real manifolds of solutions (the positive dimensional case) with corresponding advantages in applications. The stable and practical computation of real radicals in the approximate case is an important open problem. Theoretical advances and corresponding implemented algorithms are made for this problem. The approach of the thesis, is to use semidefinite programming (SDP) methods from algebraic geometry, and also techniques originating in the geometry of differential equations. The problem of finding the real radical is re-formulated as solving an SDP problem. This approach in the 00 dimensional case, was pioneered by Curto \& Fialkow with breakthroughs in the 00 dimensional case by Lasserre and collaborators. In the positive dimensional case, important contributions have been made of Ma, Wang and Zhi. The real radical corresponds to a generic point lying on the intersection of boundary of the convex cone of semidefinite matrices and a linear affine space associated with the polynomial system. As posed, this problem is not stable, since an arbitrarily small perturbation takes the point to an infeasible one outside the cone. A contribution of the thesis, is to show how to apply facial reduction pioneered by Borwein and Wolkowicz, to this problem. It is regularized by mapping the point to one which is strictly on the interior of another convex region, the minimal face of the cone. Then a strictly feasible point on the minimal face can be computed by accurate iterative methods such as the Douglas-Rachford method. Such a point corresponds to a generic point (max rank solution) of the SDP feasible problem. The regularization is done by solving the auxiliary problem which can be done again by iterative methods. This process is proved to be stable under some assumptions in this thesis as the max rank doesn\u27t change under sufficiently small perturbations. This well-posedness is also reflected in our examples, which are executed much more accurately than by methods based on interior point approaches. For a given polynomial system, and an integer d3˘e0d \u3e 0, Results of Curto \& Fialkow and Lasserre are generalized to give an algorithm for computing the real radical up to degree dd. Using this truncated real radical as input to critical point methods, can lead in many cases to validation of the real radical

    International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

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    The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    Echelon Forms and Reformulations of Pathological Semidefinite Programs

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    This thesis seeks to better understand when and why various well-known pathologies arise in semidefinite programs (SDP). The first part of this thesis is concerned with the pathology of weak infeasibility. Unlike in linear programs, Farka's lemma may fail to identify infeasible SDPs. This pathology occurs precisely when an SDP has no feasible solution, but it has nearly feasible solutions that approximate the constraint set to arbitrary precision. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to ``bad'' linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe a simple echelon form of weakly infeasible SDPs with the following properties: it is obtained by elementary row operations and congruence transformations; it makes weak infeasibility evident; and using it we can construct any weakly infeasible SDP or bad linear projection by an elementary combinatorial algorithm. We also prove that some SDPs in the literature are in our echelon form, for example, the SDP from the sum-of-squares relaxation of minimizing the famous Motzkin polynomial. The second part of this thesis deals with the pathology of exponentially sized solutions in SDPs. As a classic example of Khachiyan shows, some SDPs have solutions whose size - the number of bits necessary to describe it - is exponential in the size of the input. Although the common consent seems that large solutions in SDPs are rare, we prove that they are actually quite common: a linear change of variables transforms every strictly feasible SDP into a Khachiyan type SDP, in which the leading variables are large. As to how large, that depends on the singularity degree of a dual problem. We further show some SDPs in which large solutions appear naturally, without any change of variables. Along the way, we draw connections to continued fractions, Fourier-Motzkin elimination, and give positive progress to answering the long-standing open question: can we decide feasibility of SDPs in polynomial time?Doctor of Philosoph

    PREPROCESSING AND FIRST-ORDER PRIMAL-DUAL ALGORITHMS FOR CONVEX OPTIMIZATION

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    This thesis focuses on two topics in the field of convex optimization: preprocessing algorithms for semidefinite programs (SDPs), and first-order primal-dual algorithms for convex-concave saddle- point problems. In the first part of this thesis, we introduce Sieve-SDP, a simple facial reduction algorithm to preprocess SDPs. Sieve-SDP inspects the constraints of the problem to detect lack of strict feasibility, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on several problem collections from the literature, with many SDPs coming from polynomial optimization. In the second part, we develop two first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems involving non-bilinear coupling function, which includes SDP as one of the many special cases. Both algorithms are single-loop and have low per-iteration complexity. Our first algorithm can achieve O(1/k) convergence rates on the duality gap in both ergodic (averaging) sense and semi-ergodic sense, i.e., non-ergodic (last-iterate) on the primal, and ergodic on the dual. This rate can be further improved on non-ergodic primal objective residual using a new parameter update rule. Under strong convexity assumption, our second algorithm can boost these convergence rates to no slower than O(1/k^2). Our results can be specified to handle general convex cone constrained problems. We test our algorithms on applications such as two-player games and image processing to compare our algorithms with existing methods.Doctor of Philosoph
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