Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when s...
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Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples.
We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interp...
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We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples.
The trust-region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite-programming (SDP) relaxation. Intersecting the feasible set with a second ...
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The trust-region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite-programming (SDP) relaxation. Intersecting the feasible set with a second ellipsoid results in the two-trust-region subproblem (TTRS). Even though TTRS can also be solved in polynomial time, existing algorithms do not use SDP. In this paper, we investigate the use of SDP for TTRS. Starting from the basic SDP relaxation of TTRS, which admits a gap, recent research has tightened the basic relaxation using valid second-order-cone inequalities. Even still, closing the gap requires more. For the special case of TTRS in dimension n = 2, we fully characterize the remaining valid inequalities, which can be viewed as strengthened versions of the second-order-cone inequalities just mentioned. We also demonstrate that these valid inequalities can be used computationally even when n > 2 to solve TTRS instances that were previously unsolved using SDP-based techniques.
In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we f...
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In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvexquadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with "single" (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand's Lorentz positivity.
We consider the algorithm by Ferson et al. (Reliable computing 11(3), p. 207-233, 2005) designed for solving the NP-hard problem of computing the maximal sample variance over interval data, motivated by robust statist...
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We consider the algorithm by Ferson et al. (Reliable computing 11(3), p. 207-233, 2005) designed for solving the NP-hard problem of computing the maximal sample variance over interval data, motivated by robust statistics (in fact, the formulation can be written as a nonconvexquadratic program with a specific structure). First, we propose a new version of the algorithm improving its original time bound O(n(2).2(omega)) to O(nlogn+n & sdot;2(omega)), where n is number of input data and omega is the clique number in a certain intersection graph. Then we treat input data as random variables as it is usual in statistics) and introduce a natural probabilistic data generating model. We get 2(omega)=O(n(1/log logn)) on average. This results in average computing time O(n(1+& varepsilon;)) for & varepsilon;>0 arbitrarily small, which may be considered as "surprisingly good" average time complexity for solving an NP-hard problem. Moreover, we prove the following tail bound on the distribution of computation time: hard instances, forcing the algorithm to compute in time 2 Omega(n), occur rarely, with probability tending to zero at the rate e(-n log log n). The main result admits a smoothed-complexity interpretation: the average computing time can be bounded by n(1+O(1/sigma))/(log log n), where sigma measures the dispersion of the distribution of data perturbation.
Problems of maximizing or minimizing monotonic functions of n variables under monotonic constraints are discussed. A general framework for monotonic optimization is presented in which a key role is given to a property...
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Problems of maximizing or minimizing monotonic functions of n variables under monotonic constraints are discussed. A general framework for monotonic optimization is presented in which a key role is given to a property analogous to the separation property of convex sets. The approach is applicable to a wide class of optimization problems, including optimization problems dealing with functions representable as differences of increasing functions (d.i. functions).
We propose a new (interior) approach for the general quadraticprogramming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying t...
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We propose a new (interior) approach for the general quadraticprogramming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential.
The author (1992, 1993) earlier studied the equivalence of a class of 0-1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconv...
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The author (1992, 1993) earlier studied the equivalence of a class of 0-1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconvexquadratic programs. In this paper, a necessary and sufficient condition for local minima of this class of nonconvexquadratic programs is given;this will be the foundation for study of algorithms.
This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, they lack optimization techniques that guarant...
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This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, they lack optimization techniques that guarantee global optimality in a general setting. In this paper, we show that a class of nonconvex learning problems are equivalent to general quadratic programs. This equivalence facilitates us in developing mixed integer linear programming reformulations, which admit finite algorithms that find a provably global optimal solution. We refer to this reformulation-based technique as the mixed integer programming-based global optimization (MIPGO). To our knowledge, this is the first global optimization scheme with a theoretical guarantee for folded concave penalized nonconvex learning with the SCAD penalty [J. Amer. Statist. Assoc. 96 (2001) 1348-1360] and the MCP penalty [Ann. Statist. 38 (2001) 894-942]. Numerical results indicate a significant outperformance of MIPGO over the state-of-the-art solution scheme, local linear approximation and other alternative solution techniques in literature in terms of solution quality.
A counterexample is given to show that a previously proposed sufficient condition for a local minimum of a class of nonconvexquadratic programs is not correct. This class of problems arises in combinatorial optimizat...
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A counterexample is given to show that a previously proposed sufficient condition for a local minimum of a class of nonconvexquadratic programs is not correct. This class of problems arises in combinatorial optimization. The problem with the original proof is pointed out. (C) 1998 The Mathematical Progamming Society, Inc. Published by Elsevier Science B.V.
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