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On Augmented Lagrangian methods with general lowerlevel constraints
, 2005
"... Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. In ..."
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Cited by 84 (7 self)
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Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. Inexact resolution of the lowerlevel constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot. All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lowerlevel set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.
Grasp analysis as linear matrix inequality problems,”
 IEEE Trans. on Robotics and Automation,
, 2000
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Augmented Lagrangian methods under the Constant Positive Linear Dependence constraint qualification
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Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Cited by 31 (3 self)
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
On the convergence of augmented Lagrangian methods for constrained global optimization
 SIAM J. Optim
"... We analyze the local convergence rate of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functi ..."
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Cited by 25 (9 self)
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We analyze the local convergence rate of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold c>0. Key words: The augmented Lagrangian method, nonlinear semidefinite programming, rate of convergence, variational analysis.
Distributed rate allocation for inelastic flows: Optimization frameworks, optimality conditions, and optimal algorithms
 Purdue University, West
, 2005
"... Abstract—A common assumption behind the recent surge in research activities on network utility maximization is that the traffic flows are elastic, which implies that the utility functions are concave and there are no hard limits on the rate allocated to each flow. These critical assumptions lead to ..."
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Cited by 22 (4 self)
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Abstract—A common assumption behind the recent surge in research activities on network utility maximization is that the traffic flows are elastic, which implies that the utility functions are concave and there are no hard limits on the rate allocated to each flow. These critical assumptions lead to the tractability of the analytic models of utility maximization, but also limits the applicability of the resulting rate allocation protocols. This paper focuses on inelastic flows and removes these restrictive and often invalid assumptions. We present several optimization frameworks, optimality conditions, and optimal algorithms. First we consider nonconcave utility functions, which turn utility maximization into nonconvex, constrained optimization problems that are wellknown to be extremely difficult. We first show a surprising result that under certain conditions, the standard pricing algorithm for rate allocation will still converge to the globally optimal rate allocation. When the existing distributed algorithm fails, we present a new algorithm that produces the globally optimal rate allocation, with the worst case complexity being polynomial time in the number of users but exponential time in the number of links. In the second part of the paper, we provide a general problem formulation of rate allocation among timesensitive flows from realtime and streaming applications, as well as a decomposition into subproblems coordinated by pricing. After simplifying the subproblems by leveraging the optimization structures, we highlight the difficult issues of causality and timescale, and propose an effective pricingbased heuristics for admission control and an optimal algorithm for a special case formulation.
LOCAL CONVERGENCE OF EXACT AND INEXACT AUGMENTED LAGRANGIAN METHODS UNDER THE SECONDORDER SUFFICIENT OPTIMALITY CONDITION
, 2012
"... We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the secondorder sufficient optimality condition. In particular, no constraint qualifications of any kind ..."
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Cited by 16 (5 self)
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We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the secondorder sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the secondorder sufficient condition. That said, the classical results allow the initial multiplier estimate to be far from the optimal one, at the expense of proportionally increasing the threshold value for the penalty parameters. Although our primary goal is to avoid constraint qualifications, if the stronger assumptions are introduced, then starting points far from the optimal multiplier are allowed within our analysis as well. Using only the secondorder sufficient optimality condition, for penalty parameters large enough we prove primaldual Qlinear convergence rate, which becomes superlinear if the parameters are allowed to go to infinity. Both exact and inexact solutions of subproblems are considered. In the exact case, we further show that the primal convergence rate is of the same Qorder as the primaldual rate. Previous assertions for the primal sequence all had to do with the weaker Rrate of convergence and required the stronger assumptions cited above. Finally, we show that under our assumptions one of the popular rules of controlling the penalty parameters ensures their boundedness.
Decreasing Functions With Applications To Penalization
 SIAM J. Optimization
, 1997
"... : The theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions. A multiplicative version of the infconvolution operation is studied for decreasing functions. Modified penalty functions for some constrained optimization pro ..."
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Cited by 15 (9 self)
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: The theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions. A multiplicative version of the infconvolution operation is studied for decreasing functions. Modified penalty functions for some constrained optimization problems are introduced which are in general nonlinear with respect to the objective function of the original problem. As the perturbation function of a constrained optimization problem is decreasing, the theory of decreasing functions is subsequently applied to the study of modified penalty functions, the zero duality gap property and the exact penalization. Key Words: Decreasing functions, IPH functions, multiplicative infconvolution, modified penalty functions, exact penalization. AMS Subject Classification 1991: 90C30, 65K05 Abbreviated title: Decreasing functions and penalization 1 Introduction In this paper we study positive decreasing functions defined on the positive orthant IR n ...