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31
A quadratically convergent Newton method for computing the nearest correlation matrix
 SIAM J. Matrix Anal. Appl
, 2006
"... The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and q ..."
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Cited by 56 (14 self)
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The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasiNewton methods like BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newtontype method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method. AMS subject classifications. 49M45, 90C25, 90C33 1
ObjectiveDerivativeFree Methods for Constrained Optimization
 Mathematical Programming
, 1999
"... We propose feasible descent methods for constrained minimization that do not make explicit use of objective derivative information. The methods at each iteration sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective ..."
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Cited by 28 (8 self)
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We propose feasible descent methods for constrained minimization that do not make explicit use of objective derivative information. The methods at each iteration sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective function value is not sampled. The search arcs are obtained by projecting search direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone of firstorder feasible variations at the current iterate. We show that these methods have desirable convergence properties under certain regularity assumptions on the constraints. In the case of linear constraints, the projections are redundant and the regularity assumptions hold automatically. Numerical experience with the methods in the linear constraint case is reported. Key words. Constrained optimization, derivativefree method, feasible descent, stationary point, metric regularit...
Necessary and sufficient conditions for stable conjugate duality
"... The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ′ −φ∗(0, v), whenever a regularity condition on φ is satisfied, is a key result in convex analysis and optimization, where φ: X × Y → IR ∪ {+∞} is a convex function, X and Y are Banach spaces, Y ′ is the continuous dual space of Y a ..."
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Cited by 17 (2 self)
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The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ′ −φ∗(0, v), whenever a regularity condition on φ is satisfied, is a key result in convex analysis and optimization, where φ: X × Y → IR ∪ {+∞} is a convex function, X and Y are Banach spaces, Y ′ is the continuous dual space of Y and φ ∗ is the FenchelMoreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf x∈X {φ(x, 0) + x∗(x)} = max v∈Y ′ {−φ∗(−x∗, v)}, ∀x ∗ ∈ X ′, and obtain a new global dual regularity condition, which is much more general than the popularly known interiorpoint type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable FenchelRockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interiorpoint conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.
The SECQ, Linear Regularity and the Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces
"... We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP a ..."
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Cited by 14 (5 self)
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We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP and Jameson's (G)property) are established. With suitable continuity assumption we show how this property can be ensured from the corresponding property of some of its finite subfamilies.
Constraint qualification, the strong CHIP, and best approximation with convex constraints in Banach spaces
 SIAM J. Optim
"... Abstract. Several fundamental concepts such as the basic constraint qualification (BCQ), the strong conical hull intersection property (CHIP), and the perturbations for convex systems of inequalities in Banach spaces (over R or C) are extended and studied; here the systems are not necessarily finite ..."
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Cited by 13 (9 self)
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Abstract. Several fundamental concepts such as the basic constraint qualification (BCQ), the strong conical hull intersection property (CHIP), and the perturbations for convex systems of inequalities in Banach spaces (over R or C) are extended and studied; here the systems are not necessarily finite. Their relationships with each other in connection with the best approximations are investigated. As applications, we establish results on the unconstrained reformulation of best approximations with infinitely many constraints in Hilbert spaces; also we give several characterizations of best restricted range approximations in C(Q) under quite general constraints.
Regularities and Their Relations to Error Bounds
"... In this paper, we mainly study various notions of regularity for a finite collection {C1, · · · , Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υ ..."
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Cited by 11 (1 self)
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In this paper, we mainly study various notions of regularity for a finite collection {C1, · · · , Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υerror bound) if and only if the pair {epi(f), X ×{0}} of sets in the product space X × R is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C1, · · · , Cm} is linearly regular if and only if it has the strong CHIP and the collection {NC1(z), · · · , NCm(z)} of normal cones at z has property (G) for each z ∈ C: = ∩ m i=1Ci. Provided that C1 is a closed convex cone and that C2 = Y is a closed vector subspace of X, we show that {C1, Y} is linearly regular if and only if there exists α> 0 such that each positive (relative to the order induced by C1) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by α. Similar characterization is given in terms of normal cones.
LINEAR REGULARITY FOR A COLLECTION OF SUBSMOOTH SETS IN BANACH SPACES
"... Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover the sharpest linear r ..."
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Cited by 11 (5 self)
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Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption.
On constraint qualification for infinite system of convex inequalities in a Banach space
 SIAM J. Optim
"... Abstract. For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian–Fromovitz constraint qualification, as well as ..."
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Cited by 10 (6 self)
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Abstract. For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian–Fromovitz constraint qualification, as well as the Pshenichnyi–Levin–Valadier property introduced by Li, Nahak, and Singer. Applications are given in the restricted range approximation problem, constrained optimization problems, as well as in the approximation problem with constraints by conditionally positive semidefinite functions. Key words. system of convex inequalities, basic constraint qualification, Slater condition, MFCQ, PLV property, optimality condition, best approximation