Spelling suggestions: "subject:"constrained equation"" "subject:"nonstrained equation""
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Debugging Equation-Based Languages in OpenModelica EnvironmentSjöholm, Klas January 2009 (has links)
<p>The need for debugging tools for declarative programming languages has increased due to the rapid development of modeling and simulation tools/programs. Declarative equation-based programming languages have the problem of equation systems being over-, or under-constrained. This means that the system of equations has more equations than variables or more variables than equations respectively, making the system of equations unsolvable. In this study a static debugger is implemented in OpenModelica compiler for the equation-based programming language Modelica to make it easier for the programmer or modeler to locate the equation/s causing the unconstrained system of equations. The debugging techniques used by the debugger are developed by Peter Bunus. Those techniques are able to detect unconstrained systems of equations and give solutions by identifying the minimal set ofequation/s that should be removed or which variable/s should be added to an equation/s to make the system solvable. In this study the debugging techniques for detecting and giving a solution for over-constrained system of equations are shown suitable to be used for the programming language Modelica in the OpenModelica compiler.</p>
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Debugging Equation-Based Languages in OpenModelica EnvironmentSjöholm, Klas January 2009 (has links)
The need for debugging tools for declarative programming languages has increased due to the rapid development of modeling and simulation tools/programs. Declarative equation-based programming languages have the problem of equation systems being over-, or under-constrained. This means that the system of equations has more equations than variables or more variables than equations respectively, making the system of equations unsolvable. In this study a static debugger is implemented in OpenModelica compiler for the equation-based programming language Modelica to make it easier for the programmer or modeler to locate the equation/s causing the unconstrained system of equations. The debugging techniques used by the debugger are developed by Peter Bunus. Those techniques are able to detect unconstrained systems of equations and give solutions by identifying the minimal set ofequation/s that should be removed or which variable/s should be added to an equation/s to make the system solvable. In this study the debugging techniques for detecting and giving a solution for over-constrained system of equations are shown suitable to be used for the programming language Modelica in the OpenModelica compiler.
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Local Convergence of Newton-type Methods for Nonsmooth Constrained Equations and ApplicationsHerrich, Markus 16 January 2015 (has links) (PDF)
In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions.
The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions.
As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution.
We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions.
Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth.
Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.
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Local Convergence of Newton-type Methods for Nonsmooth Constrained Equations and ApplicationsHerrich, Markus 15 December 2014 (has links)
In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions.
The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions.
As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution.
We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions.
Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth.
Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.
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