R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), pp. 231-252.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....method that we had intended including in GALAHAD [18] despite having produced both e ective active set and interior point QP solvers. Our experience has been that without QP truncation, the cost of the QP solution so dominates that other non SQP approaches (such as IPOPT [33] KNITRO [4] and LOQO [32]) in which truncation is possible, have made signi cant progress even before our QP code had solved its rst subproblem see also [23] for further evidence that interior point methods appear to scale better than SQP ones. We are more enthusiastic about an SLP QP approach we are currently ....

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231-252, 1999.

....programming. The methods proposed in [1, 7, 12, 13] either start with a feasible point or apply a phase one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms [5, 10, 20, 23] which may enter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack based infeasible methods into feasible methods. In this framework, feasible and infeasible interior algorithms can be considered as variants of the ....

.... method for solving (2:3) that computes steps by applying Newton s method in the variables x; s; h ; g to the system i s i = i 2 I h(x) 0 g(x) Gamma s = 0; which is equivalent to the KKT conditions for (2:3) This system is the basis for primaldual infeasible algorithms; see e.g. [5, 20]. Application of Newton s method gives rise to the linear system B B xx L 0 A h (x) A g (x) 0 0 GammaS C C A 0 B B C C = Gamma B B Gammae S g C C ; 3.9) where S and denote diagonal matrices with s and g on their respective diagonals, and L stands for the ....

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R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

....D(x k ) 2 (A (2.27) D(x k ) 2 c k ) #D(x . 2.28) Again, as in the inverse barrier function, here the second term in the right hand side also has the di#erent sign from the standard system (2.3) 2.7. A transformed log barrier function method Vanderbei and Shanno[9] transforms the inequality constrained problem (1.9) 1.10) into y = 0 (2.30) 0, 2.31) by adding the slack variables y . Using the log barrier penalty to the inequality constraints (2.31) for the above problem, we obtain that log(y i ) 2.32) The first order conditions for ....

....by k 0 0 # 1 # 2 k I k 0 I # # # #y # y k k y k , 2.38) where # k = Diag( # k ) 1 , # k ) 2 , # k ) m ) 2.39) It follows from relation (2.38) that 1 # 2 # 1 # 1 e . 2. 40) In the original method of Vanderbei and Shanno[9], 2.35) is replaced by # i y i # 1 = 0, the linear system becomes k # 1 k , 2.41) where # k is the diagonal matrix defined by # k = Diag[ y k ) 1 , y k ) 2 , y k ) m ] 2.42) More details can be found also in [8] 2.8. A#ne Scaling interior point method In ....

R.J. Vanderbei and D.F. Shanno, An interior point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl. 13(1999) 231-252.

....Our testing will be done primarily on small, medium sized and moderately large problems from the CUTE collection. By this we mean problems that have up to 10,000 variables or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,28,29], but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [28] and KNITRO [5] which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point ....

....variables or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,28,29] but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [28] and KNITRO [5] which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point approaches: LOQO is a line search algorithm that has much in common with interior algorithms for linear and convex quadratic programming, ....

[Article contains additional citation context not shown here]

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

....Consider the problem s.t. c j (x) 0, j = 1, m e (P ) where f : c j : j = 1, m e and d j : j = 1, m i are smooth. No convexity assumptions are made. A number of primaldual interior point methods have been proposed to tackle such problems; see, e.g. [1, 2, 3, 4, 5, 6, 7, 8]. While all of these methods make use of a search direction generated by a Newton or quasi Newton iteration on a perturbed version of some first order necessary conditions of optimality, they di#er in many respects. For example, some algorithms enforce feasibility of all iterates with respects to ....

....necessary conditions of optimality, they di#er in many respects. For example, some algorithms enforce feasibility of all iterates with respects to inequality constraints [4, 5] while others, sometimes referred to as infeasible , sidestep that requirement via the introduction of slack variables [1, 2, 3, 6, 7, 8]. As for equality constraints, some schemes include them as is in the perturbed optimality conditions [1, 2, 3, 4, 6, 7] while some soften this condition by making use of two sets of slack variables [8] or by introducing a quadratic penalty function, yielding optimality conditions involving a ....

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R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

....can converge to a singular non stationary point even if all the iterates are feasible. Example 3 Consider the one variable problem min(x 1) 3 x subject to x 0, 4.3) whose only optimal solution is x, 0. We use a primal dual line search interior method to solve this problem. This amounts [1, 6, 9, 14] to applying the Newton iteration (1.1) to the perturbed KKT conditions for (4.3) which are given by F(x, k) x 1)2 l k) O) 4.4) x) p 0 17 where ) is the Lagrange multiplier. Let us choose the barrier parameter as = 0.01. After some algebraic manipulations one can show that the ....

....(4.9b) If the coefficient matrix in (4. 9a) is positive definite, then the search direction can be shown to be a descent direction for a variety of merit functions including b, otherwise, there are practical ways to modify it so that its eigenvalues are bounded above and away from zero for all k [8, 9, 14]. We will assume here that such a matrix modification is performed. This approach is thus similar to that used in unconstrained optimization, where global convergence can be proved under standard assumptions [5, 12] if the modified Hessian approximations have eigenvalues bounded above and away ....

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231-252, 1999.

....programming. The methods proposed in [1, 7, 12, 13] either start with a feasible point or apply a phase one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms [5, 10, 20, 23] whichmayenter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack based infeasible methods into feasible methods. In this framework, feasible and infeasible interior algorithms can be considered as variants of the same ....

.... feasible method for solving (2:3) that computes steps by applying Newton s method in the variables x# s# h# g to the system g = i s i = # i 2I ; s = 0# which is equivalent to the KKT conditions for (2:3) This system is the basis for primaldual infeasible algorithms# see e.g. [5, 20]. Application of Newton s method gives rise to the linear system B B xx L 0 A h 0 0 ;S C C A 0 B B C C = B B L(x# ) S C C (3.9) where S and denote diagonal matrices with s g on their respective diagonals, and L stands for the Lagrangian of ....

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R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

.... exact Hessian; feasibility restoration [17] and [18] knitro Trust region interior point algorithm; step decomposed into normal and tangential step; special rules for trust region adjustment [5] loqo Primal dual interior point method; line search to induce global convergence; exact Hessian [28]. snopt SQP line search method; augmented Lagrangian merit function; quasiNewton Hessian, inertia control; elastic mode for inconsistent QP subproblems. 5.1 Failures of NLP solvers All NLP solvers make some kind of constraint quali cation assumption in order to guarantee convergence. In the ....

Vanderbei, R.J. and Shanno, D.F. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231-252, 1999. A Test problem characteristics This section gives details of the characteristics all MacMPEC test problems. In the

....are still a very open topic of research in nonlinear programming. One of the issues is guaranteeing global convergence because there seems to be no ideal merit function. Several approaches for globalizing interior point methods using different merit functions have been proposed. See the references [5,8,10,15,16,22]. On the other hand, the local convergence properties of interior point methods for nonlinear programming are quite well studied in the literature [6,7,10,18,23,28] although difficulties arise when the limit point does not satisfy strict complementarity or linear independence of the gradients of ....

R. J. VANDERBEI AND D. F. SHANNO, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. and Appl., 13 (1999), pp. 231--252.

....constrained optimization. It uses a bound constrained augmented Lagrangian method. In general, LANCELOT is recommended for large problems with many degrees of freedom. It complements SNOPT and the other methods discussed above. A comparison between LANCELOT and MINOS has been made in [8, 9] LOQO [77] and KNITRO [16, 15] are examples of large scale optimization packages that treat inequality constraints by a primal dual interior method. Both packages require second derivatives but can accommodate many degrees of freedom. 1.5. Notation. Some important quantities follow: x; s) Primal, dual ....

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), pp. 231-252. Computational optimization|a tribute to Olvi Mangasarian, Part II.

.... also known as barrier methods, had an explosive development in the last fteen years, due to the success of interior point methods for linear programming and for linear complementarity problems (see the monographs [18, 9, 23, 26, 17, 25, 30] see also the extensions to nonlinear programming in [10, 27, 13, 14, 8, 3]) The rst deep study of the path of optimizers, now known as central path, is due to Bayer and Lagarias [2] and to Megiddo [20] who gave a denitive characterization of the primal dual central path. An introduction to path following methods is given by Gonzaga [15] We know that for linear ....

Vanderbei, R. J., Shanno, D. F. (1997): An interior-point algorithm for nonconvex nonlinear programming. Tech. Rep. SOR-97-21, Statistics and Operations Research, Princeton University, Princeton, USA, to appear in Computational Optimization and Applications

....Our testing will be done primarily on small, medium sized and moderately large problems from the CUTE collection. By this we mean problems that have up to 10,000 variables or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,25,26], but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [25] and KNITRO [5] which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point ....

....variables or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,25,26] but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [25] and KNITRO [5] which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point approaches: LOQO is a line search algorithm that has much in common with interior algorithms for linear and convex quadratic programming, ....

[Article contains additional citation context not shown here]

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

....of the number of active constraints. To reduce this fixed cost to only one KKT factorization per NLP interation, it becomes advantageous to develop the barrier method at the NLP level. Recently efficient barrier NLP solvers have been developed, including the LOQO solver of Vanderbei and Shanno [47] and the NITRO solver of Byrd et al. 15] For these methods there are still some limitations on convergence properties, due to nonconvex NLPs. NLP methods based on interior point concepts allow us to exploit directly all of the features mentioned above for dynamic systems. Examples that ....

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Technical Report SOR-97-21, CEOR, Princeton University, Princeton, NJ, 1997.

....to very arbitrary feasible points of MPEC. For example, there does not exist a proof that the Herskovits algorithm [20] satisfies AGP, but its application to MPEC seems to be successful [21] We conjecture that the same is true for other interior NLP algorithms as the ones described in [8] and [37]. We think that it is worthwhile to study 14 this aspect of NLP algorithms, not only from the point of view of their applicability to MPEC but also as an additional theoretical index of efficiency of NLP methods. Further research on this subject should be expected in the near future. ....

R. J. Vanderbei and D. F. Shanno [1999], An interior point algorithm for nonconvex nonlinear programming, Computational Optimization and Applications 13, pp. 231-252.

....of this subproblem with an active set quadratic programming method can be inecient and interior point (IP) or barrier) methods serve as a desirable alternative to eliminate the combinatorial problem of selecting an active set. IP techniques have been applied to the solution of large NLP problems [12, 43, 25, 38] and stem from the classical work on penalty barrier functions [22] In these approaches inequality constraints are replaced by logarithmic barrier terms in the objective function and the resulting equality constrained problem is solved with Newton type methods applied to the optimality ....

R. J. Vanderbei and D. F. Shanno, An interior point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, CEOR, Princeton University, Princeton, NJ, 1997.

....related linear least squares problems. In contrast with the algorithm presented in [1] our tilting parameter starts out positive and asymptotically approaches zero. There has been a great deal of interest recently in interior point algorithms for nonconvex nonlinear programming (see, e.g. [5, 6, 26, 4, 18, 25]) Such algorithms generate feasible iterates and typically require only the solution of linear systems of equations in order to generate new iterates. SQP type algorithms, however, are often at an advantage over such methods in the context of applications where the number of variables is not too ....

R. J. Vanderbei and D. F. Shanno, An interior point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), pp. 231--252.

....point solvers. Because of this fixed cost, it was found [26] that interior point QP solvers were not competitive with active set strategies if only a few inequality constraints become active. More recently, interior point methods have been applied directly to the solution of large NLP problems [8, 31, 16, 27] rather than the quadratic subproblem as in [26] These methods stem from the classical work on penalty functions [14] but careful analyses and modern implementations have shown that these methods are much better conditioned than previously believed [30] In these approaches inequality constraints ....

R. J. Vanderbei and D. F. Shanno, An interior point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, CEOR, Princeton University, Princeton, NJ, 1997.

....software has frequently uncovered deficiencies in the software and has generally led to software improvements. Although Mittelmann s efforts have gained the most notice, other researchers have been concerned with the evaluation and performance of optimization codes. As recent examples, we cite [1, 2, 3, 4, 6, 12, 17]. The interpretation and analysis of the data generated by the benchmarking process are the main technical issues addressed in this paper. Most benchmarking efforts involve tables displaying the performance of each solver on each problem for a set of metrics such as CPU time, number of function ....

.... Division, Argonne National Laboratory, Argonne, Illinois 60439 (dolan mcs.anl.gov) z Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439 (more mcs.anl.gov) 1 To address the shortcomings of the previous approach, some researchers rank the solvers [4, 6, 15, 17]. In other words, they count the number of times that a solver comes in kth place, usually for k = 1; 2; 3. Ranking the solvers performance for each problem helps prevent a minority of the problems from unduly influencing the results. Information on the size of the improvement, however, is lost. ....

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comp. Optim. Appl., 13 (1999), pp. 231--252. 13

....programming. The methods proposed in [1, 7, 12, 13] either start with a feasible point or apply a phase one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms [5, 10, 19, 22] which may enter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack based infeasible methods into feasible methods. In this framework, feasible and infeasible interior algorithms can be considered as variants of the ....

.... steps by applying Newton s method in the variables x; s; h ; g to the system rf(x) Gamma A h h Gamma A g g = 0 i s i = i 2 I h(x) 0 g(x) Gamma s = 0; which is equivalent to the KKT conditions for (2:3) This system is the basis for primaldual infeasible algorithms; see e.g. [5, 19]. Application of Newton s method gives rise to the linear system 0 B B r 2 xx L 0 A h (x) A g (x) 0 0 GammaS A h (x) T 0 0 0 A g (x) T GammaI 0 0 1 C C A 0 B B d x d s Gammad h Gammad g 1 C C A = Gamma 0 B B r x L(x; Gammae S g h(x) g(x) ....

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R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

....only local solutions. LOQO was not able to solve the problem for most n v 10 that we tried with default parameters. Although we found that LOQO s performance improved slightly when setting mufactor parameter small enough ( 10 Gamma4 ) which is a scale factor for the barrier parameter (see [16] and [14] We did not include the times for small n v s in the table 2.2 since all solvers took under 0:1 of a second to solve the problem. Graphics of (global) solutions for several n v are shown in the figure 2.1. 3 0.4 0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 n v = 6 0.5 0 0.5 0 0.2 0.4 0.6 ....

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, Princeton University, Princeton, New Jersey 08544, 1997. 55

....optimization problems are well understood and commercial tools are available for solving problems with thousands of variables and constraints. However, due to the relatively small problem size that needs to be solved by the resource manager, I used a freely available software tool called LOQO [102, 103]. It is also possible to write a hand tuned solver for this speci c problem, but owing to the satisfactory speed a orded by LOQO, I did not adopt this approach. In the following subsection I show a simple example that demonstrates the resource allocations computed by the resource manager after ....

R. Vanderbei and D. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231-252, 1999.

....an optimal bang bang control which is especially remarkable in the distributed control example for which we are not aware of another instance in the literature. The problems are formulated as AMPL [13] scripts and several optimization codes are applied. In particular, the interior point code LOQO [29] successfully and eciently solved all problems. It needs to be stressed that more nonlinear formulations of these or other applications can be treated in the same way. The code LOQO was designed to solve general nonconvex NLP problems. For a comparison with other codes on several classes of ....

....suitable for this purpose. Subsequently, a number of solvers, written in di erent programming languages, may be called through an interface which, in the case of AMPL, is provided for free. Below the following solvers will be used: LANCELOT A [12] MINOS 5.5 [26] SNOPT 5.34 [14] and LOQO 4. 01 [29]. Especially, the only interior point code LOQO proved to be robust and ecient for the type of problems considered. An important feature of AMPL is its automatic di erentiation capability. Only function values need to be provided for objective and constraint functions. 8 4 Numerical examples We ....

[Article contains additional citation context not shown here]

R. S. Vanderbei and D. F. Shanno, \An interior point algorithm for nonconvex nonlinear programming", to appear in Computational Optimization and Applications. 23

....problems. As was already done in [16, 18] the control problems are written in the form of AMPL [7] scripts. This way, a number of nonlinear optimization codes can be utilized for their solution. It had been an observation in our previous work that from the currently available codes only LOQO [27] is able to solve all the problems e ectively and for suciently ne discretizations. The following is independent of the solver used. Sucient Optimality for Discretized Control Problems 7 After computing a solution an AMPL stub (or :nl) le is written as well as a le with the computed Lagrange ....

R.J. Vanderbei and D.F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comp. Optim. Appl., 13, (2000), 231-252.

....related linear least squares problems. In contrast with the algorithm presented in [1] our tilting parameter starts out positive and asymptotically approaches zero. Recently there has been a great deal of interest in interior point algorithms for nonconvex nonlinear programming (see, e.g. [5, 6, 24, 4, 16, 23]) Such algorithms generate feasible iterates and typically only require the solution of linear systems of equations in order to generate new iterates. SQP type algorithms, however, are often at an advantage over such methods in the context of applications where the number of variables is not too ....

R. J. Vanderbei and D. F. Shanno. An interior point algorithm for nonconvex nonlinear programming. Technical Report SOR-97-21, Princeton University, Dept. of Statistics and Operations Research, 1997.

....points of some measure of the constraint violation, when applied to a well posed problem. 1 Introduction Over the past decade a variety of interior point methods for nonconvex nonlinear programming (NLP) have been proposed and found to be efficient in practice (see e.g. 1] 4] 6] 8] [10] [12] Based on earlier work [5] these methods come in different varieties, such as primal or primal dual methods, line search or trust region methods, with different merit functions, different strategies to update the barrier parameter, etc. For some algorithms, theoretical global convergence ....

.... under certain assumptions the considered method converges to a local solution (or at least a stationary point) of the problem from a given arbitrary starting point (see for example [1, 4, 11, 12] However, using a simple analytical example we will demonstrate that some of the current methods ([3, 4, 7, 10, 11, 12]) in particular many line search methods, can fail to converge to feasible points when applied to a well posed problem. In those cases, the limit points of the sequence of iterates can depend arbitrarily on the choice of the starting point. E mail: andreasw andrew.cmu.edu y E mail: ....

[Article contains additional citation context not shown here]

R. J. Vanderbei and D. F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. 10

....are still a very open topic of research in nonlinear programming. One of the issues is guaranteeing global convergence because there seems to be no ideal merit function. Several approaches for globalizing interior point methods using different merit functions have been proposed. See the references [4,7,8,13,14,20]. On the other hand, the local convergence properties of interior point methods for nonlinear programming are quite well studied in the literature [5,6,8,16,21,24] although difficulties arise when the limit point does not satisfy strict complementarity or linear independence of the gradients of ....

R. J. VANDERBEI AND D. F. SHANNO, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. and Appl., 13 (1999), pp. 231--252.

....Problem is too large LOQO was not able to solve the problem for most n v 10 that we tried with default parameters. However, we found that LOQO s performance improved slightly when setting the mufactor parameter small enough ( 10 Gamma4 ) which is a scale factor for the barrier parameter [14, 15]. 0.4 0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 n = 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n = 100 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n = 20 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n 10 Figure 2.1: Unit diameter polygons of maximal area 4 3 Distribution of Electrons on a Sphere (Vanderbei [13] Given n p electrons, find ....

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, Princeton University, 1997. To appear in Comp. Optim. Appl. 43

....problems. As was already done in [12] 14] the control problems are written in the form of AMPL [6] scripts. This way, a number of nonlinear optimization codes can be utilized for their solution. It had been an observation in our previous work 7 that from the currently available codes only LOQO [20] is able to solve all the problems e ectively and for suciently ne discretizations. The following is independent of the solver used. After computing a solution an AMPL stub (or :nl) le is written as well as a le with the computed Lagrange multipliers. This allows to check the SSC (4.3) with ....

R.J. Vanderbei and D.F. Shanno, \An interior-point algorithm for nonconvex nonlinear programming", Comp. Optim. Appl., vol. 13, pp. 231-252, 1999. 16

....are still a very open topic of research in nonlinear programming. One of the issues is guaranteeing global convergence because there seems to be no ideal merit function. Several approaches for globalizing interior point methods using different merit functions have been proposed. See the references [4,7,8,13,14,20]. On the other hand, the local convergence properties of interior point methods for nonlinear programming are quite well studied in the literature [5,6,8,16,21,24] although difficulties arise when the limit point does not satisfy strict complementarity or linear independence of the gradients of ....

R. J. VANDERBEI AND D. F. SHANNO, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. and Appl., 13 (1999), pp. 231--252.

....d y ; d z ) 4.5) decreases a merit function and satisfies s 0, z 0. In (4:4) the n Theta n matrix W denotes the Hessian, with respect to x, of the Lagrangian function L(x; y; z) f(x) Gamma y T c E (x) Gamma z T c I (x) 4. 6) Many interior methods follow this basic scheme [1, 13, 17, 28, 31]; they differ mainly in the choice of the merit function, in the mechanism for decreasing the barrier parameter , and in the way of handling nonconvexities. A careful implementation of a line search interior method is provided in the LOQO software package [28] This line search approach is ....

....this basic scheme [1, 13, 17, 28, 31] they differ mainly in the choice of the merit function, in the mechanism for decreasing the barrier parameter , and in the way of handling nonconvexities. A careful implementation of a line search interior method is provided in the LOQO software package [28]. This line search approach is appealing due to its simplicity and its close connection to interior methods for linear programming, which are well developed. Numerical results reported, for example, in [11, 28] indicate that these methods represent a very promising approach for solving large ....

[Article contains additional citation context not shown here]

R. J. Vanderbei and D. F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Technical Report SOR-97-21, Statistics and Operations Research, Princeton University, 1997.

.... [28] The application of these algorithms to discretized optimal control problems has also been subject of study in the papers by Battermann and Heinkenschloss [2] Leibfritz and Sachs [19] Vicente [26] and Wright [27] The recent papers by Gay, Overton, and Wright [16] and Vanderbei and Shanno [25] introduce and test globalization strategies for primal dual interior point algorithms. In the papers cited above, the step direction for the interior point method is de ned in the primal variables, in the multipliers corresponding to equality constraints and in the multipliers corresponding to ....

R. J. Vanderbei and D. F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Technical Report SOR-97-21, Statistics and Operations Research, Princeton University, 1997.

....2.2) The Exp Barrier can also be used to solve the feasibility problem for convex programming: nd with c n ( 0 8n = 1 : N (18) In fact, we will see in Section 4.3 that AdaBoost exploits that property. Solving the feasibility problem with interior point methods is more involved [48]. In the rest of the paper we concentrate on the Exp Barrier only and focus on its connection to Boosting methods. Most of the work in Section 5 can be extended to other barrier functions. 3.3 Convergence Besides an intuitive reasoning for the barrier function converging to an optimal solution ....

R.J. Vanderbei and D.F. Shanno. An interior point algorithm for nonconvex nonlinear porgramming. Technical Report SOR-97-21, Statistics and Operations Research,, Princeton University, 1997.

....s Problem is too large LOQO was not able to solve the problem for most n v 10 that we tried with default parameters. However, we found that LOQO s performance improved slightly when setting the mufactor parameter small enough ( 10 Gamma4 ) which is a scale factor for the barrier parameter [14, 15]. 0.4 0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 n = 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n = 100 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n = 20 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 n 10 Figure 2.1: Unit diameter polygons of maximal area 4 3 Distribution of Electrons on a Sphere (Vanderbei [13] Given n p electrons, find ....

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, Princeton University, 1997. To appear in Comp. Optim. Appl. 43

....trouble spots for an interior point method and propose ways to overcome them. In Section 4, we present numerical results on the MacMPEC test suite [8] 2. LOQO: An Infeasible Interior Point Method We begin with an overview of the loqo algorithm. A more detailed explanation can be found in [9]. The basic problem we consider is (6) subject to h i (x) 0, i = 1, m, where f(x) and the h i (x) s are assumed to be twice continuously differentiable and x is an n vector of otherwise unconstrained (i.e. free) variables. We add nonnegative slacks, w, to the inequality constraint ....

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231-- 252, 1999.

....of the state of the art codes available for solving large scale NLPs. Our goal is to identify features of these codes that are e#cient and those that are ine#cient on a variety of problem classes. We have worked with three algorithms: loqo, an interior point method code by Vanderbei et al. [15] . knitro, a trust region algorithm by Byrd et al. 2] snopt, a quasi Newton algorithm by Gill et al. 9] A fourth code, lancelot by Conn et al. 4] is designed for largescale nonlinear optimization, but previous work with the code [6] has shown it not competitive with the above codes ....

....The default in loqo is Multiple Minimum Degree. The details of these heuristics are given in [12] loqo also provides a mechanism for assigning priorities in factoring the columns corresponding to #x or #y. These are called, respectively, primal and dual orderings in loqo. They are discussed in [15]. loqo generally solves a modification of (7) in which #I is added to H(x, y) For # = 0, we have the Newton directions. As # # #, the directions approach the steepest descent directions. Choosing # so that H(x, y) A(x) W 1 Y A(x) #I is positive definite ensures that the step directions ....

[Article contains additional citation context not shown here]

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. Hande Y. Benson, Princeton University, Princeton, NJ David F. Shanno, Rutgers University, New Brunswick, NJ Robert J. Vanderbei, Princeton University, Princeton, NJ

.... elements presents a formidable computational challenge, the far field version of which we propose to address through formulation of a convex optimization problem in particular a second order cone program [1] followed by its numerical solution with an ultra efficient nonlinear optimizer, LOQO [2, 3], that is based on a so called infeasible interior point method. This paper develops linear and quadratic performance measures that can be used in efficient formulation of design constraints in such an optimization setting and illustrates the concepts through a simple LOQO optimized example. The ....

R.J. Vanderbei and D.F. Shanno, "An interior-point algorithm for nonconvex nonlinear programming," Tech. Rep. SOR-97-21, Statistics and Operations Research, Princeton University, 1997, To appear in Computational Optimization and Applications.

....quadratic programming algorithm. In this paper, we analyze possible ways to implement a filter based approach in an interior point algorithm. Extensive numerical testing shows that such an approach is more e#cient than using a merit function alone. 1. Introduction In three recent papers, [11], 8] and [6] the authors describe an infeasible, primal dual, interior point algorithm for nonconvex nonlinear programming and provide numerical results using the code loqo, which implements the algorithm. With nonlinear programming, it is generally impossible to use a two phased approach ....

....we have the Newton directions. As # # #, the directions approach the steepest descent directions. Choosing # so that 4 HANDE Y. BENSON, DAVID F. SHANNO, AND ROBERT J. VANDERBEI H(x, y) A(x) T W 1 Y A(x) #I is positive definite ensures that the step directions are descent directions. See [11] for further details. Having computed step directions, #x, #w, and #y, loqo then proceeds to a new point by x (k 1) x (k) # (k) #x (k) w (k 1) w (k) # (k) #w (k) 7) y (k 1) y (k) # (k) #y (k) where # (k) is chosen to ensure that w (k 1) 0, y ....

[Article contains additional citation context not shown here]

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. Hande Y. Benson, Princeton University, Princeton, NJ David F. Shanno, Rutgers University, New Brunswick, NJ Robert J. Vanderbei, Princeton University, Princeton, NJ

....of the state of the art codes available for solving large scale NLPs. Our goal is to identify features of these codes that are e#cient and those that are ine#cient on a variety of problem classes. We have worked with three algorithms: loqo, an interior point method code by Vanderbei et al. [15] . knitro, a trust region algorithm by Byrd et al. 1] snopt, a quasi Newton algorithm by Gill et al. 8] A fourth code, lancelot by Conn et al. 3] is designed for largescale nonlinear optimization, but previous work with the code [5] has shown it not competitive with the above codes ....

....The default in loqo is Multiple Minimum Degree. The details of these heuristics are given in [12] loqo also provides a mechanism for assigning priorities in factoring the columns corresponding to #x or #y. These are called, respectively, primal and dual orderings in loqo. They are discussed in [15]. loqo generally solves a modification of (7) in which #I is added to H(x, y) For # = 0, we have the Newton directions. As # # #, the directions approach the steepest descent directions. Choosing # so that H(x, y) A(x) T W 1 Y A(x) #I is positive definite ensures that the step ....

[Article contains additional citation context not shown here]

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. Hande Y. Benson, Princeton University, Princeton, NJ David F. Shanno, Rutgers University, New Brunswick, NJ Robert J. Vanderbei, Princeton University, Princeton, NJ 24 HANDE Y. BENSON, DAVID F. SHANNO, AND ROBERT J. VANDERBEI

....a line search interior point code, with SNOPT, a sequential quadraticprogramming code, and NITRO, a trust region interior point code on a large test set of nonlinear programming problems. Specific types of problems which can cause LOQO to fail are identified. 1. INTRODUCTION In two recent papers, [18], 14] a primal dual interior point algorithm for solving nonconvex nonlinear programming problems is described, together with a limited amount of computational experience with LOQO, the computer implementation of the algorithm. This algorithm is primarily a line search algorithm, although it does ....

....algorithm is primarily a line search algorithm, although it does occasionally perturb the diagonal of the Hessian matrix to ensure that the search direction is a descent direction for the merit function, a property employed by many pure trust region methods. In the preliminary testing reported in [18], 14] the algorithm showed promise of being both an efficient and robust code for general nonconvex nonlinear programming. Date: August 28, 2000. Key words and phrases. interior point methods, nonconvex optimization, nonlinear programming, jamming. Research of the first and third authors ....

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. 1

....of structures) Throughout the paper we present several optimization models. We express these models in the AMPL modeling language [10] This language provides a common mechanism for conveying problems to codes to solve them. When solving problems we generally use two different solvers: a) LOQO [18, 19, 20, 2], which implements an interior point method for general nonlinear optimization and (b) SNOPT [11] which implements an active set strategy with a quasi Newton method for the QP subproblem. This paper is intended to be a tutorial on trajectory optimization. We direct the interested reader to John ....

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. 2

....one can say that failures are most likely due simply to bad models. We express our optimization models in the AMPL modeling language [4] This language provides a common mechanism for conveying problems to codes to solve them. When solving problems we generally use two different solvers: a) LOQO [7, 8, 10, 2], which implements an interior point method for general nonlinear optimization and (b) SNOPT [5] which implements an active set strategy with a quasi Newton method for the QP subproblem. One of the lessons to be learned with the putting example is how easy it is to make a wrong model. With this ....

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999. 1 ROBERT J. VANDERBEI, PRINCETON UNIVERSITY, PRINCETON, NJ

.... elements presents a formidable computational challenge, the far field version of which we propose to address through formulation of a convex optimization problem in particular a second order cone program [1] followed by its numerical solution with an ultra efficient nonlinear optimizer, LOQO [2, 3], that is based on a so called infeasible interior point method. This paper develops linear and quadratic performance measures that can be used in efficient formulation of design constraints in such an optimization setting and illustrates the concepts through a simple LOQO optimized example. The ....

R.J. Vanderbei and D.F. Shanno, "An interior-point algorithm for nonconvex nonlinear programming," Tech. Rep. SOR-97-21, Statistics and Operations Research, Princeton University, 1997, To appear in Computational Optimization and Applications.

No context found.

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), pp. 231-252.

No context found.

R.J. Vanderbei and D.F. Shanno. An interior point algorithm for nonconvex nonlinear porgramming. Technical Report SOR-97-21, Statistics and Operations Research,, Princeton University, 1997.

No context found.

R.J. Vanderbei and D.F. Shanno. An interior point algorithm for nonconvex nonlinear porgramming. Technical Report SOR-97-21, Statistics and Operations Research,, Princeton University, 1997.

No context found.

R.J. Vanderbei, D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Tecnical report SOR-97-21, Statistics and Operations Research, Princeton University, 1997.

No context found.

R. J. Vanderbei and D. F. Shanno [1997]: An interior point algorithm for nonconvex nonlinear programming, Technical Report SOR-97-21, Statistics and Operations Research, Princeton University. 24

No context found.

R.J. Vanderbei and D.F. Shanno. An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13:231--252, 1999.

No context found.

R.J. VANDERBEI and D.F. SHANNO. An interior point algorithm for nonconvex nonlinear programming. Report SOR-97-21, Dept. of Statistics and OR, Princeton University, 1997.

No context found.

- Vanderbei, R. J., Shanno, D. F. 1999: An interior point algorithm for nonconvex nonlinear programming. Conputational Optimization and Applications, volume 13, pp. 231--252.

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