33 research outputs found
Stat Optim Inf Comput
In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to |-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.CC999999/ImCDC/Intramural CDC HHSUnited States/2022-01-01T00:00:00Z34141814PMC820532010747vault:3716
Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework
We present a new framework for the solution of mathematical programs with
equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is
viewed as a concentration of an unconstrained optimization which minimizes the
complementarity measure and a nonlinear programming with general constraints. A
strategy generalizing ideas of Byrd-Omojokun's trust region method is used to
compute steps. By penalizing the tangential constraints into the objective
function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like
strategy is used to balance the improvements on feasibility and optimality. We
show that, under MPEC-MFCQ, if the algorithm does not terminate in finite
steps, then at least one accumulation point of the iterates sequence is an
S-stationary point
Mixed-integer linearity in nonlinear optimization: a trust region approach
Bringing together nonlinear optimization with mixed-integer linear
constraints enables versatile modeling, but poses significant computational
challenges. We investigate a method to solve these problems based on sequential
mixed-integer linearization with trust region safeguard, computing feasible
iterates via calls to a generic mixed-integer linear solver. Convergence to
critical, possibly suboptimal, feasible points is established for arbitrary
starting points. Finally, we present numerical applications in nonsmooth
optimal control and optimal network design and operation.Comment: 17 pages, 3 figures, 2 table
Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming
Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context, quasi-Newton algorithms compute low-rank updates of the matrix associated with the Newton systems, instead of computing it from scratch at every iteration. In this work, we show that a simplified quasi-Newton primal-dual interior point algorithm for linear programming enjoys polynomial worst-case iteration complexity. Feasible and infeasible cases of the algorithm are considered and the most common neighborhoods of the central path are analyzed. To the best of our knowledge, this is the first attempt to deliver polynomial worst-case iteration complexity bounds for these methods. Unsurprisingly, the worst-case complexity results obtained when quasi-Newton directions are used are worse than their counterparts when Newton directions are employed. However, quasi-Newton updates are very attractive for large-scale optimization problems where the cost of factorizing the matrices is much higher than the cost of solving linear systems
Priv Stat Databases
In this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using \u2113| or \u2113| norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the \u2113|-CTA using Pseudo-Huber function was introduced in an attempt to combine positive characteristics of both \u2113|-CTA and \u2113|-CTA. All three models can be solved using appropriate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic optimization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and \u2113|-CTA as Second-Order Cone (SOC) optimization problems and test the validity of the approach on the small example of two-dimensional tabular data set.CC999999/ImCDC/Intramural CDC HHS/United States2019-11-19T00:00:00Z31745540PMC6863437693
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Optimization and Applications
Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research
Exact penalty method for D-stationary point of nonlinear optimization
We consider the nonlinear optimization problem with least -norm
measure of constraint violations and introduce the concepts of the D-stationary
point, the DL-stationary point and the DZ-stationary point with the help of
exact penalty function. If the stationary point is feasible, they correspond to
the Fritz-John stationary point, the KKT stationary point and the singular
stationary point, respectively. In order to show the usefulness of the new
stationary points, we propose a new exact penalty sequential quadratic
programming (SQP) method with inner and outer iterations and analyze its global
and local convergence. The proposed method admits convergence to a D-stationary
point and rapid infeasibility detection without driving the penalty parameter
to zero, which demonstrates the commentary given in [SIAM J. Optim., 20 (2010),
2281--2299] and can be thought to be a supplement of the theory of nonlinear
optimization on rapid detection of infeasibility. Some illustrative examples
and preliminary numerical results demonstrate that the proposed method is
robust and efficient in solving infeasible nonlinear problems and a degenerate
problem without LICQ in the literature.Comment: 24 page
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 , but has worse dependence on other numerical parameters