Let the DFP algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets, where each line search finds the first local minimum. it is prove...
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Let the DFP algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets, where each line search finds the first local minimum. it is proved that the calculated gradients are not bounded away from zero if there are only two variables. The new feature of this work is that there is no need for the objective function to be convex.
Based on the nearest-point projection of geometric convexity we give a general projection approach for solving the feasibility problem of linear programming. Application of Shor's method of space dilation gives ri...
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Based on the nearest-point projection of geometric convexity we give a general projection approach for solving the feasibility problem of linear programming. Application of Shor's method of space dilation gives rise to a family of polynomial-time ellipsoidal algorithms with improved termination criteria in case of infeasibility. Moreover, the approach renders possible application of various techniques from nonlinear programming. In particular, using a variablemetric algorithm with exact line search we obtain a fast and practically well-behaving algorithm for linear programming.
In this paper, we discuss the convergence of Broyden algorithms for the functions which are non-twice differentiable, but have LC gradient. We prove that the rate of convergence of the algorithms is linear for uniform...
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In this paper, we discuss the convergence of Broyden algorithms for the functions which are non-twice differentiable, but have LC gradient. We prove that the rate of convergence of the algorithms is linear for uniformly convex functions. We also demonstrate that under some mild conditions the algorithms are superlinsarly convergent.
In this paper, we discuss the convergence of the Broyden algorithms withrevised search direction. Under some inexact line searches, we prove that the algorithms areglobally convergent for continuously differentiable f...
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In this paper, we discuss the convergence of the Broyden algorithms withrevised search direction. Under some inexact line searches, we prove that the algorithms areglobally convergent for continuously differentiable functions and the rate of convergence of thealgorithms is one-step superlinear and n-step second-order for uniformly convex objective functions.
The development of software for minimization problems is often based on a line search method. We consider line search methods that satisfy sufficient decrease and curvature conditions, and formulate the problem of det...
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The development of software for minimization problems is often based on a line search method. We consider line search methods that satisfy sufficient decrease and curvature conditions, and formulate the problem of determining a point that satisfies these two conditions in terms of finding a point in a set T(mu). We describe a search algorithm for this problem that produces a sequence of iterates that converge to a point in T(mu) and that, except for pathological cases, terminates in a finite number of steps. Numerical results for an implementation of the search algorithm on a set of test functions show that the algorithm terminates within a small number of iterations.
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