FIX: lcp_lemke: Fix bug in 1st iterate

Author: oyamad

quantecon/optimize/lcp_lemke.py

@@ -150,7 +150,7 @@ def lcp_lemke(M, q, d=None, max_iter=10**6, piv_options=PivOptions(),
         ratio = q[i] / d[i]
         if ratio <= ratio_min + piv_options.tol_ratio_diff:
             pivrow = i
-            ratio = ratio_min
+            ratio_min = ratio
 
     _pivoting(tableau, pivcol, pivrow)
     basis[pivrow], pivcol = pivcol, pivrow + n

quantecon/optimize/tests/test_lcp_lemke.py

@@ -107,3 +107,13 @@ def test_bimatrix_game(self):
 
         res = lcp_lemke(M, q)
         _assert_success(res, M, q)
+
+    def test_bug_768(self):
+        n = 10
+        seed = 0
+        rng = np.random.RandomState(seed)
+        A = rng.standard_normal((n, n))
+        M = A.T @ A + np.eye(n)
+        q = rng.standard_normal(n)
+        res = lcp_lemke(M, q)
+        _assert_success(res, M, q)