We propose an adaptive, constraint-reduced, primal-dual interior-point algorithm for convex quadratic programming with many more inequality constraints than variables. We reduce the computational effort by assembling,...
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We propose an adaptive, constraint-reduced, primal-dual interior-point algorithm for convex quadratic programming with many more inequality constraints than variables. We reduce the computational effort by assembling, instead of the exact normal-equation matrix, an approximate matrix from a well chosen index set which includes indices of constraints that seem to be most critical. Starting with a large portion of the constraints, our proposed scheme excludes more unnecessary constraints at later iterations. We provide proofs for the global convergence and the quadratic local convergence rate of an affine-scaling variant. Numerical experiments on random problems, on a data-fitting problem, and on a problem in array pattern synthesis show the effectiveness of the constraint reduction in decreasing the time per iteration without significantly affecting the number of iterations. We note that a similar constraint-reduction approach can be applied to algorithms of Mehrotra's predictor-corrector type, although no convergence theory is supplied.
In this article, we aim to solve high-dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality, and inequality constraints. To solve the targeted QP problem to a d...
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In this article, we aim to solve high-dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality, and inequality constraints. To solve the targeted QP problem to a desired accuracy efficiently, we consider the restricted-Wolfe dual problem and develop a two-phase Proximal Augmented Lagrangian method (QPPAL), with Phase I to generate a reasonably good initial point to warm start Phase II to obtain an accurate solution efficiently. More specifically, in Phase I, based on the recently developed symmetric Gauss-Seidel (sGS) decomposition technique, we design a novel sGS-based semi-proximal augmented Lagrangian method for the purpose of finding a solution of low to medium accuracy. Then, in Phase II, a proximal augmented Lagrangian algorithm is proposed to obtain a more accurate solution efficiently. Extensive numerical results evaluating the performance of QPPAL against existing state-of-the-art solvers Gurobi, OSQP, and QPALM are presented to demonstrate the high efficiency and robustness of our proposed algorithm for solving various classes of large-scale convex QP problems. The MATLAB implementation of the software package QPPAL is available at https://***/mattohkc/softwares/qppal/.
This paper presents an algorithm for the numerical solution of constrained parameter optimization problems. The solution strategy is based on a sequential quadraticprogramming (SQP) technique that uses the L-infinity...
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This paper presents an algorithm for the numerical solution of constrained parameter optimization problems. The solution strategy is based on a sequential quadraticprogramming (SQP) technique that uses the L-infinity exact penalty function. Unlike similar SQP algorithms the method proposed here solves only strictly convexquadratic programs to obtain the search directions. The global convergence properties of the algorithm are enhanced by the use of a nonmonotone line search and second-order corrections to avoid the Maratos effect. The paper also presents an ANSI C implementation of the algorithm. The effectiveness of the proposed method is demonstrated by solving numerous parameter optimization and optimal control problems that have appeared in the literature. (C) 2007 Elsevier Inc. All rights reserved.
In this paper we give a global convergence proof of the second-order affine scaling algorithm for convex quadratic programming problems, where the new iterate is the point which minimizes the objective function over t...
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In this paper we give a global convergence proof of the second-order affine scaling algorithm for convex quadratic programming problems, where the new iterate is the point which minimizes the objective function over the intersection of the feasible region with the ellipsoid centered at the current point and whose radius is a fixed fraction beta is an element of (0;1] of the radius of the largest "scaled" ellipsoid inscribed in the nonnegative orthant. The analysis is based on the local Karmarkar potential function introduced by Tsuchiya. For any beta is an element of (0;1) and without assuming any nondegeneracy assumption on the problem, it is shown that the sequences of primal iterates and dual estimates converge to optimal solutions of the quadratic program and its dual, respectively.
This paper proposes an arc-search interior-point algorithm for convex quadratic programming with box constraints. The problem has many applications, such as optimal control with actuator saturation. It is shown that a...
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This paper proposes an arc-search interior-point algorithm for convex quadratic programming with box constraints. The problem has many applications, such as optimal control with actuator saturation. It is shown that an explicit feasible starting point exists for this problem. Therefore, an efficient feasible interior-point algorithm is proposed to tackle the problem. It is proved that the proposed algorithm is polynomial and has the best known complexity bound O (root nlog(1/is an element of). Moreover, the search neighborhood for this special problem is wider than an algorithm for general convex quadratic programming problems, which implies that longer steps and faster convergence are expected. Finally, an engineering design problem is considered and the algorithm is applied to solve the engineering problem.
In this paper we propose a new iterative method for solving a class of linear complementarity problems: u greater-than-or-equal-to 0, Mu + q greater-than-or-equal-to 0, u(T)(Mu + q) = 0, where M is a given l x l posit...
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In this paper we propose a new iterative method for solving a class of linear complementarity problems: u greater-than-or-equal-to 0, Mu + q greater-than-or-equal-to 0, u(T)(Mu + q) = 0, where M is a given l x l positive semidefinite matrix (not necessarily symmetric) and q is a given l-vector. The method makes two matrix-vector multiplications and a trivial projection onto the nonnegative orthant at each iteration, and the Euclidean distance of the iterates to the solution set monotonously converges to zero. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. It is pointed out that the method may be used to solve general convex quadratic programming problems. Preliminary numerical experiments indicate that this method may be very efficient for large sparse problems.
We consider a class of infeasible, path-following methods for convex quadratric programming. Our methods are designed to be effective for solving both nondegerate and degenerate problems, where degeneracy is understoo...
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We consider a class of infeasible, path-following methods for convex quadratric programming. Our methods are designed to be effective for solving both nondegerate and degenerate problems, where degeneracy is understood to mean the failure of strict complementarity at a solution. Global convergence and a polynomial bound on the number of iterations required is given. An implementation, CQP, is available as part of GALAHAD. We illustrate the advantages of our approach on the CUTEr and Maros-Meszaros test sets.
Murty's algorithm for the linear complementarity problem is generalized to solve the optimality conditions for linear and convex quadratic programming problems with both equality and inequality constraints. An imp...
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Murty's algorithm for the linear complementarity problem is generalized to solve the optimality conditions for linear and convex quadratic programming problems with both equality and inequality constraints. An implementation is suggested which provides both efficiency and tight error control. Numerical experiments as well as field tests in various applications show favorable results.
We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convexquadratic programs. This extension is based on the iterated application of the ...
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We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convexquadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln 3) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.
In this paper, we propose and analyse a new full-Newton step feasible interior point method for convex quadratic programming. The basic idea of this method is to replace a complementarity condition by a non-negative v...
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In this paper, we propose and analyse a new full-Newton step feasible interior point method for convex quadratic programming. The basic idea of this method is to replace a complementarity condition by a non-negative variable weight vector. With a zero of weight vector, the limit of the weighted path exists and satisfies the complementarity condition, the limit yields an optimal solution of problem. In each main iteration of the new algorithm consisted of only full-Newton steps with a quadratic rate of convergence. The advantage of this method is the use of a full-Newton step, that is no calculation of the step size is required. Finally, some numerical results are reported to show the practical performance of the proposed algorithm with different parameters.
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