On the global optimization properties of finite-difference local descent algorithms

The paper is devoted to the convergence properties of finite-difference local descent algorithms in global optimization problems with a special -convex structure. It is assumed that the objective function can be closely approximated by some smooth convex function. Stability properties of the perturbed gradient descent and coordinate descent methods are investigated. Basing on this results some global optimization properties of finite-difference local descent algorithms, in particular, coordinate descent method, are discovered. These properties are not inherent in methods using exact gradients.The paper was presented at the II. IIASA-Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Perturbed steepest-descent technique in multiextremal problems

The steepest-descent technique dealing with the perturbed values of the objective function and its gradients and with nonexact line searches is considered. Attention is given to the case where the perturbations do not decrease on the algorithm trajectories; the aim is to investigate how perturbations at every iteration of the algorithm perturb its original attractor set.Based on the Liapunov direct method for attraction analysis of discrete-time processes, a sharp estimation of the attractor set generated by a perturbed steepest-descent technique with respect to the perturbation magnitudes is obtained. Some global optimization properties of finite-difference analogues of the gradient method are discovered. These properties are not inherent in methods which use exact gradients.The author is grateful to the referees for many useful suggestions.     

Asymptotic properties of penalized objective functions in regular mathematical programming problems

Translated from Vychislitel'nye Kompleksy i Modelirovanie Slozhnykh Sistem, pp. 117–124, Moscow State University, 1989.     

Error Stability Properties of Generalized Gradient-Type Algorithms

We present a unified framework for convergence analysis of generalized subgradient-type algorithms in the presence of perturbations. A principal novel feature of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an -stationary set of the problem, where depends on the magnitude of perturbations. Characterization of the attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp, and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel, and heavy ball modifications.     

Finiteε-subgradient algorithm for approximate solution of the parametric programming problem

Translated from Vychislitel'nye Kompleksy i Modelirovanie Slozhnykh Sistem, pp. 111–117, Moscow State University, 1989.     

Error tolerance of the conditional gradient method

Translated from Programmnoe Obespechenie i Modeli Sistemnogo Analiza, pp. 161–165, 1991.     

Two-dye and one- or two-quencher DNA probes for real-time PCR assay: synthesis and comparison with a TaqMan? probe

A typical TaqMan? real-time PCR probe contains a 5'-fluorescent dye and a 3'-quencher. In the course of the amplification, the probe is degraded starting from the 5'-end, thus releasing fluorescent dye. Some fluorophores (including fluorescein) are known to be prone to self-quenching when located near each other. This work is aimed at studying dye-dye and dye-quencher interactions in multiply modified DNA probes. Twenty-one fluorogenic probes containing one and two fluoresceins (FAM), or a FAM-JOE pair, and one or two BHQ1 quenchers were synthesized using non-nucleoside reagents and "click chemistry" post-modification on solid phase and in solution. The probes were tested in real-time PCR using an ~300-bp-long natural DNA fragment as a template. The structural prerequisites for lowering the probe background fluorescence and increasing the end-plateau fluorescence intensity were evaluated and discussed.     

A trajectory algorithm based on the gradient method I. The search on the quasioptimal trajectories

The global optimization problem is considered under the assumption that the objective function is convex with respect to some variables. A finite subgradient algorithm for the search of an -optimal solution is proposed. Results of numerical experiments are presented.