Distributionally Robust Optimization Approach For Two Kinds Of Problems With Uncertain Data

Distributionally robust optimization approach is an effective method to reduce the risks resulting from incomplete information on the distribution of the uncertainty.In this ^trt_tddr^pohh_iissu^leebe_tte^ts _rr^uei s_iid^{s s}e_bbi^it,t _uusⁿ _ttt^ofwoo_iir^pef_ooi^{r t} _bnn^fimid^rm_{u a}i_flt^asis_rli^ttzt_oyoⁱr^aoc i_nmr^tonbⁱ _o^{n o}u_otb^osntX_nhⁿⁱi _u ^d _eo_st^tme n_ths^hro s_eeo^{e d t} _ti_p^dhe._us_t^ile _in^{ss t}c_Tcm^aaro_ehi^bisn_rzet^bsss_a^oout_att^ltcr_irin^uiiu_oa^oat c_cnd^etnt _t^ee_aa^voM_m^dafd_{tb oa}^{l twu}_{le d}b^hie _ae^ty_sln^{e he} _es^ut_d^q _m^tnh^uhc_{a itr}e_-eh^aee _{d e}^trlim_ee^te _fs^oaao_oitnpa^nissm_i^nt _{t bt}e_ile^wsdn_im^{iq a}B_st_pa^ttu _hlr^haai _{e o}n_dd^riuef_{g e}^sno_rbc^ucr_aiy^menm_sm^or _iu^ata_oms^dvat_ni^eaii_i^lin _no_nig^{la a}n_sg^{dr b} _tfb^eal w_hoa^tee._e^ah_rs^s,i_m^t _efc_id^Dawh_ur^ib _sl^s _ota^hltic^e _tn^rso_ai^{rih e}n_ont^{b e}t_ndh^tuda_s^ht,_eiⁱ _s^e n_oe^{o w}s_wfc^cn _oo^hoat_trn^elmh_hsd^lrye_et^e-moments of the random variables describing the parameters uncertainty.The numerical test also show the feasibility of the distributionally robust optimization model to solve uncertain absolute value equations.Besides,we consider the optimization problem with uncertain objective function.Distributionally robust optimization approach associated with the minimax regret model is established,which is called the distributionally robust minimax regret optimization approach.To deal with the data uncertainty,we propose a randomization method by introducing Monte-carlo sampling method and carry out the convergence analysis.By approximating the inner finite maximum function with a con-tinuous differentiable approximation function,the distributionally minimax regret model is transformed into a semi-infinite min-max-min problem and the optimality function can be developed.Then we use an adaptive smoothing technique to construct a sequence of finite min-max problems,which,together with their optimality functions are consistent approximations to the original pair.Finally,an algorithm using the min-max algorithm as the main program is proposed to deal with the semi-infinite min-max-min problem.