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Some modifications of the limited-memory variable metric optimization methods
Vlček, Jan ; Lukšan, Ladislav
Several modifications of the limited-memory variable metric (or quasi-Newton) line search methods for large scale unconstrained optimization are investigated. First the block version of the symmetric rank-one (SR1) update formula is derived in a similar way as for the block BFGS update in Vlˇcek and Lukˇsan (Numerical Algorithms 2019). The block SR1 formula is then modified to obtain an update which can reduce the required number of arithmetic operations per iteration. Since it usually violates the corresponding secant conditions, this update is combined with the shifting investigated in Vlˇcek and Lukˇsan (J. Comput. Appl. Math. 2006). Moreover, a new efficient way how to realize the limited-memory shifted BFGS method is proposed. For a class of methods based on the generalized shifted economy BFGS update, global convergence is established. A numerical comparison with the standard L-BFGS and BNS methods is given.
Plný tet:  PDF
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Two limited-memory optimization methods with minimum violation of the previous quasi-Newton equations
Vlček, Jan ; Lukšan, Ladislav
Limited-memory variable metric methods based on the well-known BFGS update are widely used for large scale optimization. The block version of the BFGS update, derived by Schnabel (1983), Hu and Storey (1991) and Vlček and Lukšan (2019), satisfies the quasi-Newton equations with all used difference vectors and for quadratic objective functions gives the best improvement of convergence in some sense, but the corresponding direction vectors are not descent directions generally. To guarantee the descent property of direction vectors and simultaneously violate the quasi-Newton equations as little as possible in some sense, two methods based on the block BFGS update are proposed. They can be advantageously combined with methods based on vector corrections for conjugacy (Vlček and Lukšan, 2015). Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new methods.
Plný tet: PDF
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Application of the Infinitely Many Times Repeated BNS Update and Conjugate Directions to Limited-Memory Optimization Methods
Vlček, Jan ; Lukšan, Ladislav
To improve the performance of the L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed. Since this can be time consuming, the extra updates need to be selected carefully. We show that groups of these updates can be repeated infinitely many times under some conditions, without a noticeable increase of the computational time. The limit update is a block BFGS update. It can be obtained by solving of some Lyapunov matrix equation whose order can be decreased by application of vector corrections for conjugacy. Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical results indicate the efficiency of the new method.
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A Generalized Limited-Memory BNS Method Based on the Block BFGS Update
Vlček, Jan ; Lukšan, Ladislav
A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in some sense for quadratic objective functions, but it does not guarantee that the direction vectors are descent for general functions. To overcome this difficulty and utilize the advantageous properties of the block BFGS update, a block version of the limited-memory BNS method for large scale unconstrained optimization is proposed. The algorithm is globally convergent for convex sufficiently smooth functions and our numerical experiments indicate its efficiency.
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Metody vnitřních bodů pro zobecněnou minimaxovou optimalizaci
Lukšan, Ladislav ; Matonoha, Ctirad ; Vlček, Jan
A new class of primal interior point methods for generalized minimax optimization is described. These methods use besides a standard logarithmic barrier function also barrier functions bounded from below which have more favourable properties for investigation of global convergence. It deals with descent direction methods, where an approxmation of the Hessian matrix is computed by gradient differences or quasi-Newton updates. Two-level optimization is used. A direction vector is computed by a Choleski decompostition of a sparse matrix. Numerical experiments concerning two basic applications, minimization of a point maximum and a sum of absolute values of smooth functions, are presented.
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