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11
An InteriorPoint Algorithm For Nonconvex Nonlinear Programming
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 1997
"... The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the mer ..."
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Cited by 144 (13 self)
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The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
Robust Truss Topology Design via Semidefinite Programming
 OPTIMIZATION LABORATORY, FACULTY OF INDUSTRIAL ENGINEERING AND MANAGEMENT, TECHNION – THE ISRAEL INSTITUTE OF TECHNOLOGY, TECHNION CITY, HAIFA 32000
, 1995
"... We present and motivate a new model of the Truss Topology Design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small “occasional” loads is optimized. It is shown that the resulting optimization problem is a Semidefinite Program. We derive and ana ..."
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Cited by 49 (8 self)
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We present and motivate a new model of the Truss Topology Design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small “occasional” loads is optimized. It is shown that the resulting optimization problem is a Semidefinite Program. We derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples.
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
"... ..."
Group symmetry in interiorpoint methods for semidefinite programming
 Optimization and Engineering
, 1970
"... Abstract A class of group symmetric SemiDefinite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of primaldual interiorpoint methods are group symmetric. Preservation of group symmetry along the se ..."
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Cited by 12 (1 self)
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Abstract A class of group symmetric SemiDefinite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of primaldual interiorpoint methods are group symmetric. Preservation of group symmetry along the search direction theoretically guarantees that the numerically obtained optimal solution is group symmetric. As an illustrative example, we show that the optimization problem of a symmetric truss under frequency constraints can be formulated as a group symmetric SDP. Numerical experiments using an interiorpoint algorithm demonstrate convergence to strictly group symmetric solutions.
Using LOQO To Solve SecondOrder Cone Programming Problems
 PRINCETON UNIVERSITY
, 1998
"... Many nonlinear optimization problems can be cast as secondorder cone programming problems. In this paper, we discuss a broad spectrum of such applications. For each application, we consider various formulations, some convex some not, and study which ones are amenable to solution using a generalpur ..."
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Cited by 11 (0 self)
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Many nonlinear optimization problems can be cast as secondorder cone programming problems. In this paper, we discuss a broad spectrum of such applications. For each application, we consider various formulations, some convex some not, and study which ones are amenable to solution using a generalpurpose interiorpoint solver LOQO. We also compare with other commonly available nonlinear programming solvers and specialpurpose codes for secondorder cone programming.
SemiDefinite Problems in Truss Topology Optimization
, 1995
"... In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class of proble ..."
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Cited by 5 (2 self)
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In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class of problems. Key words: truss topology design, positive semidefinite programming iii 1 Introduction The discrete truss topology design (TTD) problem is to find optimal volumes of bars connecting a set of nodes, which may be free or fixed. The bar volumes may be zero, thus capturing the combinatorial nature of the problem, namely which bars to include in an optimal design. The nodes may be chosen as a fixed dense grid (ground structure), which allows a degree of optimization of nodal positions in the final design as well [16, 3]. The truss structure is designed to withstand specified forces (loading scenarios). The ground structure discretization approach lends itself to the application of mathema...
Topology Optimization of MultiComponent Structures via DecompositionBased Assembly Synthesis
, 2003
"... A method is presented for synthesizing multicomponent structural assemblies with maximum structural performance and manufacturability. The problem is posed as a relaxation of decompositionbased assembly synthesis [1,2,3], where both topology and decomposition of a structure are regarded as variabl ..."
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Cited by 2 (1 self)
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A method is presented for synthesizing multicomponent structural assemblies with maximum structural performance and manufacturability. The problem is posed as a relaxation of decompositionbased assembly synthesis [1,2,3], where both topology and decomposition of a structure are regarded as variables over a ground structure with nonoverlapping beams. A multiobjective genetic algorithm [4,5] with graphbased crossover [6,7,8], coupled with FEM analyses, is used to obtain Pareto optimal solutions to this problem, exhibiting tradeoffs among structural stiffness, total weight, component manufacturability (size and simplicity), and the number of joints. Case studies with a cantilever and a simplified automotive floor frame are presented, and representative designs in the Pareto front are examined for the tradeoffs among the multiple criteria.
A.: Synthesis of Bistable Periodic Structures Using Topology Optimization and a Genetic Algorithm
 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conferences
, 2006
"... A formulation for the automatic synthesis of twodimensional bistable, compliant periodic structures is presented, based on standard methods for topology optimization. The design space is parametrized using nonlinear beam elements and a ground structure approach. A performance criterion is suggested ..."
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Cited by 2 (0 self)
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A formulation for the automatic synthesis of twodimensional bistable, compliant periodic structures is presented, based on standard methods for topology optimization. The design space is parametrized using nonlinear beam elements and a ground structure approach. A performance criterion is suggested, based on characteristics of the loaddeformation curve of the compliant structure. A genetic algorithm is used to find candidate solutions. A numerical implementation of this methodology is discussed and illustrated using simple examples. �DOI: 10.1115/1.2338576� 1
EXTREME OPTICS AND THE SEARCH FOR EARTHLIKE PLANETS
, 2006
"... Abstract. In this paper I describe a new and exciting application of optimization technology. The problem is to design a space telescope capable of imaging Earthlike planets around nearby stars. Because of limitations inherent in the wave nature of light, the design problem is one of diffraction co ..."
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Cited by 1 (0 self)
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Abstract. In this paper I describe a new and exciting application of optimization technology. The problem is to design a space telescope capable of imaging Earthlike planets around nearby stars. Because of limitations inherent in the wave nature of light, the design problem is one of diffraction control so as to provide the extremely high contrast needed to image a faint planet positioned very close to its much brighter star. I will describe the mathematics behind the diffraction control problem and explain how modern optimization tools were able to provide unexpected solutions that actually changed NASA’s approach to this problem.
InteriorPoint Methods for Nonlinear, SecondOrder Cone, and . . .
, 2001
"... Interiorpoint methods have been a reemerging field in optimization since the mid1980s. We will present here ways of improving the performance of these algorithms for nonlinear optimization and extending them to different classes of problems and application areas. At each iteration, an interiorpo ..."
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Interiorpoint methods have been a reemerging field in optimization since the mid1980s. We will present here ways of improving the performance of these algorithms for nonlinear optimization and extending them to different classes of problems and application areas. At each iteration, an interiorpoint algorithm computes a direction in which to proceed, and then must decide how long of a step to take. The traditional approach to choosing a steplength is to use a merit function, which balances the goals of improving the objective function and satisfying the constraints. Recently, Fletcher and Leyffer reported success with using a filter method, where improvement of any of the objective function and constraint infeasibility is sufficient. We have combined these two approaches and applied them to interiorpoint methods for the first time and with good results. Another issue in nonlinear optimization is the emergence of several popular problem classes and their specialized solution algorithms. Two such problem classes are SecondOrder Cone Programming (SOCP) and Semidefinite Programming (SDP). In the second part of this dissertation, we show that problems from both of these classes can be reformulated as smooth convex optimization problems and solved using a general purpose interiorpoint algorithm for nonlinear optimization.