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An FPTAS for minimizing the product of two nonnegative linear cost functions
 MATH. PROGRAMMING
"... We consider a quadratic programming (QP) problem (Π) of the form min xT Cx subject to Ax ≥ b where C ∈ Rn×n +, rank(C) = 1 and A ∈ Rm×n, b ∈ Rm. We present an FPTAS for this problem by reformulating the QP (Π) as a parametrized LP and “rounding ” the optimal solution. Furthermore, our algorithm ret ..."
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Cited by 6 (1 self)
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We consider a quadratic programming (QP) problem (Π) of the form min xT Cx subject to Ax ≥ b where C ∈ Rn×n +, rank(C) = 1 and A ∈ Rm×n, b ∈ Rm. We present an FPTAS for this problem by reformulating the QP (Π) as a parametrized LP and “rounding ” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 01 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and submodular flows. We also extend our results to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, tshortest paths and s, tmincuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear nonnegative cost functions.
Block structured quadratic programming for the direct multiple shooting method for optimal control
 Optimization Methods and Software
, 2011
"... Abstract. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting ..."
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Abstract. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but sparse quadratic programs to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for quadratic programs with many controls. This algorithm is based on a hybrid null–space range–space approach to exploit the block sparse structure of the quadratic programs that is due to direct multiple shooting. An assessment of the theoretical run time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally we demonstrate the merit of the new complementary condensing approach by comparing the behavior of both methods for a vehicle control problem in which the integer gear decision is convexified. 1.
Reliable solution of convex quadratic programs with parametric active set methods
, 2010
"... Parametric Active Set Methods (PASM) are a relatively new class of methods to solve convex Quadratic Programming (QP) problems. They are based on tracing the solution along a linear homotopy between a QP with known solution and the QP to be solved. We explicitly identify numerical challenges in PA ..."
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Cited by 4 (1 self)
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Parametric Active Set Methods (PASM) are a relatively new class of methods to solve convex Quadratic Programming (QP) problems. They are based on tracing the solution along a linear homotopy between a QP with known solution and the QP to be solved. We explicitly identify numerical challenges in PASM and develop strategies to meet these challenges. To demonstrate the reliability improvements of the proposed strategies we have implemented a prototype code called rpasm. We compare the results of the code with those of other popular academic and commercial QP solvers on problems of a subset of the MarosMészáros test examples. Our code is the only one to solve all of the problems with the default settings. Furthermore, we propose how PASM can be used to compute local solutions of nonconvex QPs.
Lowrank modification of riccati factorizations with applications to model predictive control
 In Proceedings of the 52nd IEEE Conference on Decision and Control
, 2013
"... ©2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wo ..."
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Cited by 4 (3 self)
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©2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
The generalized trust region subproblem
, 2012
"... The interval bounded generalized trust region subproblem (GTRS) consists in minimizing a general quadratic objective, q0(x) → min, subject to an upper and lower bounded general quadratic constraint, ℓ ≤ q1(x) ≤ u. This means that there are no definiteness assumptions on either quadratic function. ..."
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Cited by 2 (0 self)
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The interval bounded generalized trust region subproblem (GTRS) consists in minimizing a general quadratic objective, q0(x) → min, subject to an upper and lower bounded general quadratic constraint, ℓ ≤ q1(x) ≤ u. This means that there are no definiteness assumptions on either quadratic function. We first study characterizations of optimality for this implicitly convex problem under a constraint qualification and show that it can be assumed without loss of generality. We next classify the GTRS into easy case and hard case instances, and demonstrate that the upper and lower bounded general problem can be reduced to an equivalent equality constrained problem after identifying suitable generalized eigenvalues and possibly solving a sparse system. We then discuss how the RendlWolkowicz algorithm proposed in [11, 29] can be extended to solve the resulting equality constrained problem, highlighting the connection between the GTRS and the problem of finding minimum generalized eigenvalues of a parameterized
nonnegative linear cost functions
, 2008
"... Abstract We consider a quadratic programming (QP) problem () of the form min xT Cx subject to Ax ≥ b, x ≥ 0 where C ∈ Rn×n+, rank(C) = 1 and A ∈ R m×n, b ∈ Rm. We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP () as a parameterized LP and “rou ..."
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Abstract We consider a quadratic programming (QP) problem () of the form min xT Cx subject to Ax ≥ b, x ≥ 0 where C ∈ Rn×n+, rank(C) = 1 and A ∈ R m×n, b ∈ Rm. We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP () as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and submodular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, tshortest paths and s, tmincuts. For the above discrete problems, the quadratic program models the problem of obtaining an integer solution that minimizes the product of two linear nonnegative cost functions.
Nonlinear Feedback Neural Network for Solution of Quadratic Programming Problems
"... This paper presents a recurrent neural circuit for solving quadratic programming problems. The objective is tominimize a quadratic cost function subject to linearconstraints. The proposed circuit employs nonlinearfeedback, in the form of unipolar comparators, to introducetranscendental terms in the ..."
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This paper presents a recurrent neural circuit for solving quadratic programming problems. The objective is tominimize a quadratic cost function subject to linearconstraints. The proposed circuit employs nonlinearfeedback, in the form of unipolar comparators, to introducetranscendental terms in the energy function ensuring fastconvergence to the solution. The proof of validity of the energy function is also provided. The hardware complexity of the proposed circuit comparesfavorably with other proposed circuits for the same task. PSPICE simulation results arepresented for a chosen optimization problem and are foundto agree with the algebraic solution.