Add handling for multipliers that matches the reg_homotopy solver

[apozharski]

Also fix incorrect default sizes for solver_initialization.

[nosnoc-ci-bot]

Summary - mpecopt coverage

Summary

Generated on:	03/20/2025 - 15:50:49
Coverage date:	03/20/2025 - 15:50:31
Parser:	Cobertura
Assemblies:	2
Classes:	47
Files:	47
Line coverage:	31.5% (1184 of 3755)
Covered lines:	1184
Uncovered lines:	2571
Coverable lines:	3755
Total lines:	5942
Covered branches:	0
Total branches:	0
Method coverage:	Feature is only available for sponsors
Tag:	25_13973563551

Coverage

- 15.6%

Name	Line	Branch
****	15.6%	****
addaptive_active_set_identification	0%
BNLPProjectionStrategy
check_feasiblity_of_lpec	0%
check_if_B_stationary	0%
create_comparison_table	0%
create_lpec_functions	100%
create_lpec_subproblem	48.9%
create_mpec_functions	68.7%
create_phase_i_nlp_solver	0%
create_phase_i_nlp_solver_dev	0%
determine_multipliers_based_stationary_point	14.6%
fast_B_stationarity_check	0%
find_active_sets	100%
find_active_sets_piece_nlp	68%
find_nonscalar	90%
HomotopySolverOptions	100%
HomotopySteering
InitializationStrategy
lpec_fallback_strategy	0%
lpec_fallback_strategy_phase_ii	0%
lpec_solver	20%
LpecSolver
LPECSolverOptions	100%
mpec_active_set_generate_problem_functions	0%
mpec_active_set_nosnoc	0%
mpec_active_set_preprocess	0%
mpec_homotopy_solver	0%
mpec_optimizer	0%
mpecopt_phase_ii	0%
nice_plot_colors	0%
PieceNLPStartegy
plot_lpec	0%
print_iter_details	0%
print_iter_header	100%
print_iter_header_objective	0%
print_iter_line	100%
print_iter_line_objective	0%
print_iter_stats	91.6%
print_iter_stats_objective	0%
print_iter_summary	100%
print_iter_summary_scholtes	0%
print_phase_i	100%
print_phase_ii	100%
project_to_bounds	100%
unfold_struct	0%

mpecopt - 62.5%

Name	Line	Branch
mpecopt	62.5%	****
mpecopt.Options	100%
mpecopt.Solver	61.9%

[nurkanovic]

LGTM