A study of search directions in primal-dual interior-point methods for semidefinite programming

A study of search directions in primal-dual interior-point methods for semidefinite programmin

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Credible Autocoding of Convex Optimization Algorithms

The efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical proofs on on-line optimization programs which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming (SDP) solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-to-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks about how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be adressed in future work

Credible Autocoding of Convex Optimization Algorithms

International audienceThe efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical proofs on on-line optimization programs which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming (SDP) solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-to-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks about how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be adressed in future work

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters

Group-invariant Semidefinite Programming and Applications

Abstract This essay considers semidefinite programming problems that exhibit a special form of symmetry called group-invariance. We demonstrate the effect of such symmetries on certain path-following interior-point algorithms, and highlight a reduction technique that is particularly useful on certain groupinvariant semidefinite programming problems. Two applications of groupinvariant semidefinite programming problems-one in truss design and the other in graph theory-are presented. ii Acknowledgements To my supervisor, Dr. Chek Beng Chua, for his wise advice, great encouragement and continuous support during my master's study. To Professor Michael Best for his comments and careful reading of the draft. To my dear friends who gave me great help and made my life in Waterloo a wonderful experience

Computational analysis of real-time convex optimization for control systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2000.Includes bibliographical references (p. 177-189).Computational analysis is fundamental for certification of all real-time control software. Nevertheless, analysis of on-line optimization for control has received little attention to date. On-line software must pass rigorous standards in reliability, requiring that any embedded optimization algorithm possess predictable behavior and bounded run-time guarantees. This thesis examines the problem of certifying control systems which utilize real-time optimization. A general convex programming framework is used, to which primal-dual path-following algorithms are applied. The set of all optimization problem instances which may arise in an on-line procedure is characterized as a compact parametric set of convex programming problems. A method is given for checking the feasibility and well-posedness of this compact set of problems, providing certification that every problem instance has a solution and can be solved in finite time. The thesis then proposes several algorithm initialization methods, considering the fixed and time-varying constraint cases separately. Computational bounds are provided for both cases. In the event that the computational requirements cannot be met, several alternatives to on-line optimization are suggested. Of course, these alternatives must provide feasible solutions with minimal real-time computational overhead. Beyond this requirement, these methods approximate the optimal solution as well as possible. The methods explored include robust table look-up, functional approximation of the solution set, and ellipsoidal approximation of the constraint set. The final part of this thesis examines the coupled behavior of a receding horizon control scheme for constrained linear systems and real-time optimization. The driving requirement is to maintain closed-loop stability, feasibility and well-posedness of the optimal control problem, and bounded iterations for the optimization algorithm. A detailed analysis provides sufficient conditions for meeting these requirements. A realistic example of a small autonomous air vehicle is furnished, showing how a receding horizon control law using real-time optimization can be certified.by Lawrence Kent McGovern.Ph.D

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis

A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap.This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector