We propose a feasibledirection approach for the minimization by interior-point algorithms of a smooth function under smooth equality and inequality constraints. It consists of the iterative solution in the primal and...
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We propose a feasibledirection approach for the minimization by interior-point algorithms of a smooth function under smooth equality and inequality constraints. It consists of the iterative solution in the primal and dual variables of the Karush-Kuhn-Tucker first-order optimality conditions. At each iteration, a descent direction is defined by solving a linear system. In a second stage, the linear system is perturbed so as to deflect the descent direction and obtain a feasible descent direction. A line search is then performed to get a new interior point and ensure global convergence. Based on this approach, first-order, Newton, and quasi-Newton algorithms can be obtained. To introduce the method, we consider first the inequality constrained problem and present a globally convergent basic algorithm. Particular first-order and quasi-Newton versions of this algorithm are also stated. Then, equality constraints are included. This method, which is simple to code, does not require the solution of quadratic programs and it is neither a penalty method nor a barrier method. Several practical applications and numerical results show that our method is strong and efficient.
At each iteration, the algorithm determines a feasible descent direction by minimizing a linear or quadratic approximation to the cost on the feasible set. The algorithm is easy to implement if the approximation is ea...
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This paper studies the dynamic user optimal (DUO) traffic assignment problem considering simultaneous route and departure time choice. The DUO problem is formulated as a discrete variational inequality (DVI), with an ...
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This paper studies the dynamic user optimal (DUO) traffic assignment problem considering simultaneous route and departure time choice. The DUO problem is formulated as a discrete variational inequality (DVI), with an embeded LWR-consistent mesoscopic dynamic network loading (DNL) model to encapsulate traffic dynamics. The presented DNL model is capable of capturing realistic traffic phenomena such as queue spillback. Various VI solution algorithms, particularly those based on feasibledirections and a line search, are applied to solve the formulated DUO problem. Two examples are constructed to check equilibrium solutions obtained from numerical algorithms, to compare the performance of the algorithms, and to study the impacts of traffic interacts across multiple links on equilibrium solutions.
This paper deals with bilevel programs with strictly convex lower level problems. We present the theoretical basis of a kind of necessary and sufficient optimality conditions that involve a single-level mathematical p...
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This paper deals with bilevel programs with strictly convex lower level problems. We present the theoretical basis of a kind of necessary and sufficient optimality conditions that involve a single-level mathematical program satisfying the linear independence constraint qualification. These conditions are obtained by replacing the inner problem by their optimality conditions and relaxing their inequality constraints. An algorithm for the bilevel program, based on a well known technique for classical smooth constrained optimization, is also studied. The algorithm obtains a solution of this problem with an effort similar to that required by a classical well-behaved nonlinear constrained optimization problem. Several illustrative problems which include linear, quadratic and general nonlinear functions and constraints are solved, and very good results are obtained for all cases.
This paper presents a comparative analysis of contact algorithms used for solving contact shape optimization problems. Specifically, a nonlinear, feasibledirection interior point method (FDIPM) for the frictional con...
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This paper presents a comparative analysis of contact algorithms used for solving contact shape optimization problems. Specifically, a nonlinear, feasibledirection interior point method (FDIPM) for the frictional contact analysis of hyperelastic materials has been implemented in which the friction is introduced using the return mapping approach. This comparative investigation aims to find the cause of instability in sensitivity of the contact pressure nonuniformity (CPN). The results obtained by the FDIPM are found to be comparable with those by the penalty methods (PM) and the augmented Lagrange multiplier methods (ALMM);however, the FDIPM possesses advantages, including good convergence and convenience in modeling. Furthermore, the basic cause of the unstable sensitivity is revealed to be the discretization of the finite element method, which causes the discontinuous increase of contact area with respect to the continuous increase of contact load. To improve the stability of the CPN, an adaptive post-processing technique is proposed.
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