Implement solution saving and exporting for refactored code

[abelsiqueira]

• Update function to compute dual
• Create function to save solution and dual values to DuckDB connection
• Update save_solution_to_file
• Update tests

Related issues

Closes #115
Closes #818

Checklist

• I am following the contributing guidelines

• Tests are passing

• Lint workflow is passing

• Docs were updated and workflow is passing

[github-actions]

Benchmark Results

	8322b5c...	1285127...	8322b5c.../128512721d0c7a...
energy_problem/create_model	32.4 ± 2.1 s	32.3 ± 2.5 s	1
energy_problem/input_and_constructor	17.3 ± 0.57 s	17.9 ± 0.44 s	0.967
time_to_load	4.09 ± 0.03 s	4.2 ± 0.13 s	0.973

	8322b5c...	1285127...	8322b5c.../128512721d0c7a...
energy_problem/create_model	0.311 G allocs: 15.6 GB	0.311 G allocs: 15.6 GB	1
energy_problem/input_and_constructor	0.0532 G allocs: 1.98 GB	0.0531 G allocs: 1.97 GB	1
time_to_load	0.159 k allocs: 11.2 kB	0.159 k allocs: 11.2 kB	1

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[codecov]

Codecov Report

Attention: Patch coverage is 93.33333% with 4 lines in your changes missing coverage. Please review.

Project coverage is 95.06%. Comparing base (8322b5c) to head (1285127).

Files with missing lines	Patch %	Lines
src/solve-model.jl	91.30%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1000      +/-   ##
==========================================
- Coverage   95.43%   95.06%   -0.38%     
==========================================
  Files          29       29              
  Lines        1140     1155      +15     
==========================================
+ Hits         1088     1098      +10     
- Misses         52       57       +5     

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