In this paper, we construct a new spline smoothing homotopy method for solving general nonlinear programming problems with a large number of complicated constraints. We transform the equality constraints into the ineq...
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In this paper, we construct a new spline smoothing homotopy method for solving general nonlinear programming problems with a large number of complicated constraints. We transform the equality constraints into the inequality constraints by introducing two parameters. Subsequently, we use smooth spline functions to approximate the inequality constraints. The smooth spline functions involve only few inequality constraints. In other words, the method introduces an active set technique. Under some weaker conditions, we obtain the global convergence of the new spline smoothing homotopy method. We perform numerical tests to compare the new method to other methods, and the numerical results show that the new spline smoothing homotopy method is highly efficient.
A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It ...
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A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It is shown that the Lagrangian dual is a special case of this general dual and that other special cases turn out to be LP problems. The LP duals provide a very powerful computational device but are derived using fairly strict conditions. Hence, they can often be infeasible even if the primal NLP problem is feasible and bounded. Many directions for relaxing these conditions are outlined for future research. A concept of local duality is also introduced for the first time akin to the concept of local optimality. (C) 1999 Elsevier Science B.V. All rights reserved.
The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCo...
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The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCormick in 1967 and an improved version was given by Fiacco and McCormick in their 1968 award-winning book. Later it was learned that in 1953 Pennisi had presented exactly the same theory. Many authors, most notably Maurer and Zowe in a widely cited paper in 1978, argue that the Pennisi-McCormick theory cannot be extended to infinite dimensions without adding further assumptions, by producing a counterexample. They then extend the theory to infinite dimensions, allowing for an infinite number of constraints, by strengthening the sufficient conditions required. In the current paper we use a fundamental principle for second-order sufficiency to extend the Pennisi-McCormick second-order theory as stated in R-n to infinite-dimensional normed vector spaces, without strengthening the conditions. The Maurer and Zowe infinite-dimensional counterexample carried an infinite number of constraints. Hence they seemed to be unaware that the extension of the Pennisi-McCormick theory to infinite dimensions was possible provided the original feature of a finite number of constraints was maintained.
We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a mo...
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We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a modified Newton method and use an tau(2)-exact penalty function to attain feasibility. Our methods have strong global convergence properties under standard assumptions. Specifically, if the penalty parameter remains bounded, any limit point of the iterate sequence is either a Karush-Kuhn-Tucker (KKT) point of the barrier subproblem, or a Fritz-John (FJ) point of the original problem that fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ);if the penalty parameter tends to infinity, there is a limit point that is either an infeasible FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given that illustrate these outcomes.
The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINL...
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The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINLP) presenting a challenge even to the modern heuristic search methods. In the proposed hybrid method, The ACOA is used to determine pipe diameters while the NLP is used to determine the pipe slopes of the network by proposing two different formulations. In the first formulation, named ACOA-NLP1 a penalty method is used to satisfy the problem constraints while in the second one, named ACOA-NLP2, the velocity and flow depth constraints are expressed in terms of the slope constraints which are easily satisfied as box constraint of the NLP solver leading to a considerable reduction of the search space size. In addition, the assumption of minimum cover depth at the network inlets is used to calculate the nodal cover depths and the pump and drop heights at the network nodes, if required, leading to a complete solution. The total cost of the constructed solution is used as the objective function of the ACOA, guiding the ant toward minimum cost solutions. Proposed hybrid methods are used to solve three test examples, and the results are presented and compared with those produced by a conventional application of ACOA. The results indicate the effectiveness and efficiency of the proposed formulations and in particular the ACOA-NLP2 to optimally solve the sewer network design optimization problems. (C) 2018 Water Environment Federation
In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111-124 (2007)]. Three robust...
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In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111-124 (2007)]. Three robust nonlinear programming problems were chosen from the literature using the criteria that results calculated using other methods must be available and the problems should be realistic, but fairly simple. Our studies show that the first-order method produced reasonable solutions when the level of uncertainty was small to moderate. In addition, we demonstrate a method for leveraging a theoretical result to eliminate constraint violations. Since the first-order method is relatively inexpensive in comparison to other robust optimization techniques, our studies indicate that, under moderate uncertainty, the first-order approach may be more suitable than other methods for large problems.
This paper proposes a line search filter reduced Hessian method for nonlinear equality constrained optimization. The feature of the presented algorithm is that the reduced Hessian method is used to produce a search di...
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This paper proposes a line search filter reduced Hessian method for nonlinear equality constrained optimization. The feature of the presented algorithm is that the reduced Hessian method is used to produce a search direction, a backtracking line search procedure to generate step size, some filtered rules to determine step acceptance, second order correction technique to reduce infeasibility and overcome the Maratos effects. It is shown that this algorithm does not suffer from the Maratos effects by using second order correction step, and under mild assumptions fast convergence to second order sufficient local solutions is achieved. The numerical experiment is reported to show the effectiveness of the proposed algorithm. (C) 2011 Elsevier Inc. All rights reserved.
Automatic assignment of tolerances to dimensioned mechanical assemblies is studied as an optimization problem;the objective of which is to minimize the (manufacturing) cost, subject to the constraints of (design) func...
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Automatic assignment of tolerances to dimensioned mechanical assemblies is studied as an optimization problem;the objective of which is to minimize the (manufacturing) cost, subject to the constraints of (design) functionality and (assembly) interchangeability. By associating a nominal dimension and a tolerance to the variance, a probabilistic approach is adopted. Trigonometric functions relating the component geometries give rise to the nonlinearity in the system. Estimating an n-dimensional nonlinear integral by a polytope converts the probabilistic optimization formulation to a deterministic one. It also allows rapid evaluation of tolerance analysis embedded in tolerance synthesis. Local optimality is ensured by analysis of convexity and quasi-concavity of the objective function and some of the constraints. Sensitivity analysis is performed to provide search directions for global optimality. An implementation is reported with an example.
We construct a family of globally defined dynamical systems for a nonlinear programming problem, such that (a) the equilibrium points are the unknown (and sought) critical points of the problem, (b) for every initial ...
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We construct a family of globally defined dynamical systems for a nonlinear programming problem, such that (a) the equilibrium points are the unknown (and sought) critical points of the problem, (b) for every initial condition, the solution of the corresponding initial value problem converges to the set of critical points, (c) every strict local minimum is locally asymptotically stable, (d) the feasible set is a positively invariant set, and (e) the dynamical system is given explicitly and does not involve the unknown critical points of the problem. No convexity assumption is employed. The construction of the family of dynamical systems is based on an extension of the control Lyapunov function methodology, which employs extensions of LaSalle's theorem and are of independent interest. Examples illustrate the obtained results.
The past few years have led to the development of a novel large-scale nonlinear programming solver called IPOPT. Described in Wachter and Biegler (2004), this algorithm uses a barrier formulation for inequality constr...
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The past few years have led to the development of a novel large-scale nonlinear programming solver called IPOPT. Described in Wachter and Biegler (2004), this algorithm uses a barrier formulation for inequality constraints and incorporates a new filter line search algorithm. It also includes a number of features that make it efficient and robust even for highly nonlinear problems. In particular, it is well suited to exploit second derivatives. The code has been used on thousands on test problems ranging in size up to almost two million variables. Finally, enhancements of this method have been made to deal with complementarity conditions that model a class of discrete decisions with continuous variables. This paper describes the development of CAPE-OPEN compliant objects for IPOPT to solve dynamic optimization problems. This activity complements a number of tasks to develop interfaces to optimization modelling platforms such as AIMMS and AMPL. Here we consider protocols such as Equation Set Objects (ESO) and the MINLP CAPE-OPEN interface to IPOPT (CO-LaN Consortium, 2002). The resulting object can be linked to other objects that are CAPE-OPEN compliant. We also describe a preprocessing procedure for generic models that leads to efficient optimization problem formulations for IPOPT. To validate the interface and the preprocessing procedure, we present a comprehensive optimization case study that links to dynamic optimization models written in gPROMS. In addition, future plans for the development, enhancement and distribution of IPOPT will be outlined.
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