Use GraphOfConvexSets::SolveConvexRestriction in rounding. (#19816)

Also fixes SolveConvexRestriction to properly rewrite the convex costs
into constriants (with slack variables) to be supported by the convex
solvers. Previously it could call SNOPT instead of, e.g., Mosek.

bindings/pydrake/geometry/test/optimization_test.py

@@ -531,7 +531,7 @@ def test_graph_of_convex_sets(self):
         self.assertIsNone(options.max_rounded_paths)
         options.convex_relaxation = True
         options.preprocessing = False
-        options.max_rounded_paths = 2
+        options.max_rounded_paths = 0
         options.max_rounding_trials = 5
         options.flow_tolerance = 1e-6
         options.rounding_seed = 1

geometry/optimization/BUILD.bazel

@@ -167,7 +167,9 @@ drake_cc_library(
         "//common/symbolic:expression",
         "//solvers:choose_best_solver",
         "//solvers:create_cost",
+        "//solvers:get_program_type",
         "//solvers:mathematical_program_result",
+        "//solvers:mosek_solver",
         "//solvers:solver_interface",
     ],
 )