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A Global Optimization Method, αBB, for General TwiceDifferentiable Constrained NLPs: I  Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the constru ..."
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Cited by 52 (3 self)
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In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
A Sqp Method For General Nonlinear Programs Using Only Equality Constrained Subproblems
 MATHEMATICAL PROGRAMMING
, 1993
"... In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working s ..."
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Cited by 46 (2 self)
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In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQPmethods, as demonstrated by extensive numerical tests.
A Branch and Cut Algorithm for Nonconvex Quadratically Constrained Quadratic Programming
, 1999
"... We present a branch and cut algorithm that yields in finite time, a globally ffloptimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a ..."
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Cited by 24 (4 self)
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We present a branch and cut algorithm that yields in finite time, a globally ffloptimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using ReformulationLinearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported. Some problems of the literature are solved, for the first time with a proof of global optimality.
Global Optimization in Generalized Geometric Programming
 Engng
, 1997
"... A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints ..."
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Cited by 12 (3 self)
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A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints and the objective are decomposed into the difference of two convex functions. A convex relaxation of problem (DC) is then obtained based on the linear lower bounding of the concave parts of the objective function and constraints inside some box region. The proposed branch and bound type algorithm attains finite fflconvergence to the global minimum through the successive refinement of a convex relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of nonlinear convex optimization problems. The efficiency of the proposed approach is enhanced by eliminating variables through monotonicity analysis, by maintaining tightly bound variables thro...
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
A New Technique For Inconsistent QP Problems In The SQP Method
 University at Darmstadt, Department of Mathematics, preprint 1561, Darmstadt
, 1993
"... Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the ..."
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Cited by 7 (2 self)
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Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the simple technique of Pantoja and Mayne [34] and the highly successful, but expensive one of Tone [44]. Global convergence of a corresponding algorithm is shown under reasonable weak conditions. Numerical results are reported which show that this technique, combined with a special method for the case of regular subproblems, is quite competitive to highly appreciated established ones. Key words: sequential quadratic programming, SQP method, nonlinear programming AMS(MOS) subject classification: primary 90C30, secondary 65K05 1 NOTATION Superscripts on a vector denote elements of sequences. All vectors are column vectors. For a vectorvalued function g rg(x) denotes the transposed Jacobian eval...
Comparative Assessment of Algorithms and Software for Global Optimization
 Journal of Global Optimization
, 2005
"... The thorough evaluation of global optimization algorithms and software demands devotion, time and (hardware) resources, in addition to professional objectivity. ..."
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Cited by 3 (3 self)
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The thorough evaluation of global optimization algorithms and software demands devotion, time and (hardware) resources, in addition to professional objectivity.
Conefree” primaldual pathfollowing and potential reduction polynomial time interiorpoint methods
 Math. Prog
, 2005
"... Abstract. We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory an ..."
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Cited by 1 (1 self)
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Abstract. We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interiorpoint methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primaldual interiorpoint techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.
An Algorithm for Posynomial Geometric Programming, Based on Generalized Linear Programming
, 1995
"... This paper describes a column generation algorithm for posynomial geometric programming that ..."
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This paper describes a column generation algorithm for posynomial geometric programming that