Abstract
In this article we consider schemes of the error estimates development for numerical methods solving optimization problems. Suggested error estimates are formulated as functions depended on various values generated by numerical methods (i.e. points, function values, gradient values, etc.). These functions do not depend on such problem’s parameters like Lipshitz constant, strong convexity constant and etc. The numerical value of the error estimates is calculated after receiving accurate enough problem’s solution. Error estimates were received both for target function value and argument value.
The research was supported by Russian Science Foundation (project No. 21-71-30005).
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Notes
- 1.
Parameters of the objective function allows explicitly find problem solution \(x^* = \beta \), \(f(x^*) = 0\).
References
Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London (1981)
Gasnikov, A.V.: Modern Numerical Optimization Methods, 2nd edn. MIPT, Moscow (2018)
Nesterov, Y.E.: Introduction to Convex Optimization. MTsNMO, Moscow (2018)
Nemirovski, A.S., Iudin, D.B.: Problem Complexity and Method Efficiency in Optimization. Nauka, Moscow (1979)
Bubeck, S.: Convex optimization: algorithms and complexity. Found. Trends Mach. Learn. 8(3–4), 231–357 (2015)
Gasnikov, A.V., et al.: Accelerated meta-algorithm for convex optimization problems. Comput. Math. Math. Phys. 61(1), 17–28 (2021). https://doi.org/10.1134/S096554252101005X
Birjukov, A.G., Grinevich, A.I.: Multiprecision arithmetic as a guarantee of the required accuracy of numerical calculation results. TRUDY MIPT 4(3), 171–180 (2012)
Birjukov, A.G., Grinevich, A.I.: Methods for rounding errors estimation in the solution of numerical problems using arbitrary precision floating point calculations. TRUDY MIPT 5(2), 160–174 (2013)
Chernov, A., Dvurechensky, P., Gasnikov, A.: Fast primal-dual gradient method for strongly convex minimization problems with linear constraints. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 391–403. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_31
Anikin, A.S., Gasnikov, A.V., Dvurechensky, P.E., et al.: Dual approaches to the minimization of strongly convex functionals with a simple structure under affine constraints. Comput. Math. and Math. Phys. 57(8), 1270–1284 (2017)
Polyak, B.T.: Introduction to Optimization, 2nd edn., corr. and ext. LENAND, Moscow (2014)
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Birjukov, A., Chernov, A. (2021). On Numerical Estimates of Errors in Solving Convex Optimization Problems. In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds) Advances in Optimization and Applications. OPTIMA 2021. Communications in Computer and Information Science, vol 1514. Springer, Cham. https://doi.org/10.1007/978-3-030-92711-0_1
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