diff --git a/pyomo/contrib/piecewise/piecewise_linear_function.py b/pyomo/contrib/piecewise/piecewise_linear_function.py
index df0bbd3bf93..80fd658989a 100644
--- a/pyomo/contrib/piecewise/piecewise_linear_function.py
+++ b/pyomo/contrib/piecewise/piecewise_linear_function.py
@@ -256,6 +256,10 @@ class PiecewiseLinearFunction(Block):
             as specified, trusting the user in the case of AssumeValid.
             When no argument or None is passed, the default is
             Triangulation.Unknown
+        convex (optional): If True, serves as a user-provided guarantee that the
+            piecewise-linear function is convex. If False, serves as a user-provided
+            guarantee that the piecewise-linear function is concave. If None (the
+            default), the function may be concave or convex.
     """
 
     _ComponentDataClass = PiecewiseLinearFunctionData
@@ -283,6 +287,7 @@ def __init__(self, *args, **kwargs):
         _tabular_data_arg = kwargs.pop('tabular_data', None)
         _tabular_data_rule_arg = kwargs.pop('tabular_data_rule', None)
         _triangulation_rule_arg = kwargs.pop('triangulation', None)
+        _convex_rule_arg = kwargs.pop('convex', None)
 
         kwargs.setdefault('ctype', PiecewiseLinearFunction)
         Block.__init__(self, *args, **kwargs)
@@ -304,6 +309,9 @@ def __init__(self, *args, **kwargs):
         self._triangulation_rule = Initializer(
             _triangulation_rule_arg, treat_sequences_as_mappings=False
         )
+        self._convex_rule = Initializer(
+            _convex_rule_arg, treat_sequences_as_mappings=False
+        )
 
     def _get_dimension_from_points(self, points):
         if len(points) < 1:
@@ -606,6 +614,11 @@ def _getitem_when_not_present(self, index):
             )
         obj = handler(self, obj, parent, nonlinear_function)
 
+        # Update convexity info
+        obj.convex = None
+        if self._convex_rule is not None:
+            obj.convex = self._convex_rule(parent, index)
+
         return obj
 
 
diff --git a/pyomo/contrib/piecewise/tests/test_epigraph_hypograph.py b/pyomo/contrib/piecewise/tests/test_epigraph_hypograph.py
new file mode 100644
index 00000000000..d37a056897b
--- /dev/null
+++ b/pyomo/contrib/piecewise/tests/test_epigraph_hypograph.py
@@ -0,0 +1,335 @@
+# ____________________________________________________________________________________
+#
+# Pyomo: Python Optimization Modeling Objects
+# Copyright (c) 2008-2026 National Technology and Engineering Solutions of Sandia, LLC
+# Under the terms of Contract DE-NA0003525 with National Technology and Engineering
+# Solutions of Sandia, LLC, the U.S. Government retains certain rights in this
+# software. This software is distributed under the 3-clause BSD License.
+# ____________________________________________________________________________________
+
+import pyomo.common.unittest as unittest
+from pyomo.contrib.piecewise import PiecewiseLinearFunction
+from pyomo.contrib.piecewise.transform.epigraph_hypograph import (
+    PWLToEpigraphOrHypograph,
+)
+from pyomo.core.expr.compare import assertExpressionsEqual
+from pyomo.environ import (
+    ConcreteModel,
+    Constraint,
+    Set,
+    Var,
+    Objective,
+    TransformationFactory,
+    SolverFactory,
+    maximize,
+    minimize,
+    value,
+)
+from pyomo.contrib.piecewise.transform.piecewise_linear_transformation_base import (
+    PiecewiseLinearTransformationBase,
+)
+
+gurobi_available = (
+    SolverFactory('gurobi').available(exception_flag=False)
+    and SolverFactory('gurobi').license_is_valid()
+)
+
+
+def x_squared(x):
+    return x**2
+
+
+def neg_x_squared(x):
+    return -(x**2)
+
+
+class TestEpigraphHypographTransformation(unittest.TestCase):
+    """Test the epigraph/hypograph transformation for piecewise linear functions."""
+
+    def pwl_x_squared(self):
+        # Create a model with a convex piecewise linear function
+        m = ConcreteModel()
+        m.x = Var(bounds=(0, 4))
+
+        # Define breakpoints for x in [0, 4]
+        breakpoints = [0, 1, 2, 3, 4]
+
+        # Create piecewise linear approximation of x²
+        m.pw = PiecewiseLinearFunction(
+            points=breakpoints,
+            function=x_squared,
+            convex=True,  # Specify that this is convex
+        )
+
+        m.c = Constraint(expr=m.x >= 1.5)
+
+        # Use the piecewise function in an objective - minimize to test the piecewise part
+        m.obj = Objective(expr=m.pw(m.x), sense=minimize)
+
+        return m
+
+    def test_convex_pwl_minimization(self):
+        """Test transformation of a convex piecewise linear function (x² approximation)."""
+        m = self.pwl_x_squared()
+        # Apply the epigraph transformation
+        transformation = TransformationFactory('contrib.piecewise.epigraph_hypograph')
+        transformation.apply_to(m)
+
+        # Verify the transformation created the expected components
+        # The transformation should have created:
+        # 1. A substitute variable
+        # 2. Epigraphical constraints (one for each linear segment)
+
+        # Get the substitute var and the transformed function block
+        sub_var = m.pw.get_transformation_var(m.obj.expr)
+        self.assertIsInstance(sub_var, Var)
+        pw_block = sub_var.parent_block()
+
+        # Check that epigraphical constraints exist
+        epigraphical_constraints = pw_block.component('epigraphical_constraints')
+        self.assertIsInstance(epigraphical_constraints, Constraint)
+
+        # For 5 breakpoints, we have 4 segments, so we should have 4 constraints
+        self.assertEqual(len(epigraphical_constraints), 4)
+
+        # Verify the objective now uses the substitute variable
+        # The original pw(m.x) should be replaced with substitute_var
+        obj_expr = m.obj.expr.expr
+        self.assertIs(obj_expr, sub_var)
+
+        exprs = [
+            m.x - sub_var,
+            3.0 * m.x - 2.0 - sub_var,
+            5.0 * m.x - 6.0 - sub_var,
+            7.0 * m.x - 12.0 - sub_var,
+        ]
+
+        # Verify each constraint is of the form: linear_func_expr <= substitute_var
+        # (since this is a convex function, we use epigraph)
+        for idx, cons in epigraphical_constraints.items():
+            constraint = pw_block.epigraphical_constraints[idx]
+            # For epigraph of convex function, we expect <= constraints
+            # The constraint is stored as: linear_func_expr <= substitute_var
+            # which Pyomo represents as: linear_func_expr - substitute_var <= 0
+            # So the upper bound should be 0
+            self.assertEqual(constraint.upper, 0)
+            self.assertIsNone(constraint.lower)
+            print(constraint.body)
+            assertExpressionsEqual(self, constraint.body, exprs[idx])
+
+    @unittest.skipUnless(gurobi_available, "Gurobi is not available")
+    def test_solve_pwl_minimization(self):
+        m = self.pwl_x_squared()
+        # Apply the epigraph transformation
+        transformation = TransformationFactory('contrib.piecewise.epigraph_hypograph')
+        transformation.apply_to(m)
+
+        # Now solve the model to verify it produces the correct solution
+        # The minimum of x² for x >= 1.5 is at x = 1.5, with value 2.25
+        solver = SolverFactory('gurobi')
+        results = solver.solve(m)
+
+        # Check that the solve was successful
+        self.assertTrue(
+            results.solver.termination_condition == 'optimal'
+            or results.solver.termination_condition == 'feasible'
+        )
+
+        # Check the solution
+        # The optimal x should be 1.5 (the lower bound)
+        self.assertAlmostEqual(m.x.value, 1.5, places=4)
+
+        # The objective value should be approximately 2.25 (1.5²)
+        expected_obj = 3 * 1.5 - 2
+        self.assertAlmostEqual(value(m.obj.expr), expected_obj, places=4)
+
+    def pwl_neg_x_squared(self):
+        # Create a model with a concave piecewise linear function
+        m = ConcreteModel()
+        m.x = Var(bounds=(0, 4))
+
+        # Define breakpoints for x in [0, 4]
+        breakpoints = [0, 1, 2, 3, 4]
+
+        # Create piecewise linear approximation of -x²
+        m.pw = PiecewiseLinearFunction(
+            points=breakpoints,
+            function=neg_x_squared,
+            convex=False,  # Specify that this is concave
+        )
+
+        m.c = Constraint(expr=m.x <= 2.5)
+
+        # Use the piecewise function in an objective - maximize to test the
+        # piecewise part
+        m.obj = Objective(expr=m.pw(m.x), sense=maximize)
+
+        return m
+
+    def test_concave_pwl_maximization(self):
+        """Test transformation of a concave piecewise linear function (-x²
+        approximation)."""
+        m = self.pwl_neg_x_squared()
+        # Apply the hypograph transformation
+        transformation = TransformationFactory('contrib.piecewise.epigraph_hypograph')
+        transformation.apply_to(m)
+
+        # Verify the transformation created the expected components
+        # The transformation should have created:
+        # 1. A substitute variable
+        # 2. Epigraphical constraints (one for each linear segment)
+
+        # Get the substitute var and the transformed function block
+        sub_var = m.pw.get_transformation_var(m.obj.expr)
+        self.assertIsInstance(sub_var, Var)
+        pw_block = sub_var.parent_block()
+
+        # Check that epigraphical constraints exist
+        epigraphical_constraints = pw_block.component('epigraphical_constraints')
+        self.assertIsInstance(epigraphical_constraints, Constraint)
+
+        # For 5 breakpoints, we have 4 segments, so we should have 4 constraints
+        self.assertEqual(len(epigraphical_constraints), 4)
+
+        # Verify the objective now uses the substitute variable
+        # The original pw(m.x) should be replaced with substitute_var
+        obj_expr = m.obj.expr.expr
+        self.assertIs(obj_expr, sub_var)
+
+        exprs = [
+            sub_var - (-1.0 * m.x + 0.0),
+            sub_var - (-3.0 * m.x + 2.0),
+            sub_var - (-5.0 * m.x + 6.0),
+            sub_var - (-7.0 * m.x + 12.0),
+        ]
+
+        # Verify each constraint is of the form: linear_func_expr >= substitute_var
+        # (since this is a concave function, we use hypograph)
+        for idx, cons in epigraphical_constraints.items():
+            constraint = pw_block.epigraphical_constraints[idx]
+            # For hypograph of concave function, we expect >= constraints
+            # The constraint is stored as: linear_func_expr >= substitute_var
+            # which Pyomo represents as: substitute_var - linear_func_expr <= 0
+            # So the upper bound should be 0
+            self.assertEqual(constraint.upper, 0)
+            self.assertIsNone(constraint.lower)
+            assertExpressionsEqual(self, constraint.body, exprs[idx])
+
+    def pwl_indexed_convex_concave(self):
+        # Create a model with an IndexedPiecewiseLinearFunction with two
+        # indices: one convex (x²) and one concave (-x²). This time the
+        # piecewise linear functions are used in Constraints rather than
+        # an Objective.
+        m = ConcreteModel()
+        m.idx = Set(initialize=[1, 2])
+        m.x = Var(m.idx, bounds=(0, 4))
+
+        breakpoints = [0, 1, 2, 3, 4]
+
+        def func_rule(m, i):
+            return x_squared if i == 1 else neg_x_squared
+
+        def convex_rule(m, i):
+            return True if i == 1 else False
+
+        m.pw = PiecewiseLinearFunction(
+            m.idx, points=breakpoints, function_rule=func_rule, convex=convex_rule
+        )
+
+        m.c1 = Constraint(expr=m.pw[1](m.x[1]) <= 10)
+        m.c2 = Constraint(expr=m.pw[2](m.x[2]) >= -10)
+
+        return m
+
+    def test_indexed_convex_concave_constraints(self):
+        """Test transformation of an IndexedPiecewiseLinearFunction with one
+        convex and one concave index, used in Constraints."""
+        m = self.pwl_indexed_convex_concave()
+
+        transformation = TransformationFactory('contrib.piecewise.epigraph_hypograph')
+        transformation.apply_to(m)
+
+        # pw[1] is convex, so it gets an epigraph. pw[2] is concave, so it
+        # gets a hypograph.
+        sub_var1 = m.pw[1].get_transformation_var(m.c1.body)
+        sub_var2 = m.pw[2].get_transformation_var(m.c2.body)
+        self.assertIsInstance(sub_var1, Var)
+        self.assertIsInstance(sub_var2, Var)
+
+        # Verify the constraints now use the substitute variables
+        self.assertIs(m.c1.body.expr, sub_var1)
+        self.assertIsNone(m.c1.lower)
+        self.assertEqual(m.c1.upper, 10)
+        self.assertIs(m.c2.body.expr, sub_var2)
+        self.assertEqual(m.c2.lower, -10)
+        self.assertIsNone(m.c2.upper)
+
+        pw_block1 = sub_var1.parent_block()
+        pw_block2 = sub_var2.parent_block()
+
+        epigraphical_constraints1 = pw_block1.component('epigraphical_constraints')
+        epigraphical_constraints2 = pw_block2.component('epigraphical_constraints')
+        self.assertIsInstance(epigraphical_constraints1, Constraint)
+        self.assertIsInstance(epigraphical_constraints2, Constraint)
+        self.assertEqual(len(epigraphical_constraints1), 4)
+        self.assertEqual(len(epigraphical_constraints2), 4)
+
+        exprs1 = [
+            m.x[1] - sub_var1,
+            3.0 * m.x[1] - 2.0 - sub_var1,
+            5.0 * m.x[1] - 6.0 - sub_var1,
+            7.0 * m.x[1] - 12.0 - sub_var1,
+        ]
+        for idx, cons in epigraphical_constraints1.items():
+            self.assertEqual(cons.upper, 0)
+            self.assertIsNone(cons.lower)
+            assertExpressionsEqual(self, cons.body, exprs1[idx])
+
+        exprs2 = [
+            sub_var2 - (-1.0 * m.x[2] + 0.0),
+            sub_var2 - (-3.0 * m.x[2] + 2.0),
+            sub_var2 - (-5.0 * m.x[2] + 6.0),
+            sub_var2 - (-7.0 * m.x[2] + 12.0),
+        ]
+        for idx, cons in epigraphical_constraints2.items():
+            self.assertEqual(cons.upper, 0)
+            self.assertIsNone(cons.lower)
+            assertExpressionsEqual(self, cons.body, exprs2[idx])
+
+    def test_indexed_convex_concave_targets(self):
+        """Test that the 'targets' argument can be used to transform only
+        one of the indices of an IndexedPiecewiseLinearFunction."""
+        m = self.pwl_indexed_convex_concave()
+
+        transformation = TransformationFactory('contrib.piecewise.epigraph_hypograph')
+        transformation.apply_to(m, targets=[m.pw[1]])
+
+        # pw[1] should be transformed...
+        self.assertFalse(m.pw[1].active)
+        sub_var1 = m.pw[1].get_transformation_var(m.c1.body)
+        self.assertIsInstance(sub_var1, Var)
+        self.assertIs(m.c1.body.expr, sub_var1)
+
+        pw_block1 = sub_var1.parent_block()
+        epigraphical_constraints1 = pw_block1.component('epigraphical_constraints')
+        self.assertIsInstance(epigraphical_constraints1, Constraint)
+        self.assertEqual(len(epigraphical_constraints1), 4)
+
+        exprs1 = [
+            m.x[1] - sub_var1,
+            3.0 * m.x[1] - 2.0 - sub_var1,
+            5.0 * m.x[1] - 6.0 - sub_var1,
+            7.0 * m.x[1] - 12.0 - sub_var1,
+        ]
+        for idx, cons in epigraphical_constraints1.items():
+            self.assertEqual(cons.upper, 0)
+            self.assertIsNone(cons.lower)
+            assertExpressionsEqual(self, cons.body, exprs1[idx])
+
+        # ...but pw[2] should *not* be transformed.
+        self.assertTrue(m.pw[2].active)
+        self.assertIsNone(m.pw[2].get_transformation_var(m.c2.body))
+
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/pyomo/contrib/piecewise/tests/test_piecewise_linear_function.py b/pyomo/contrib/piecewise/tests/test_piecewise_linear_function.py
index 5bfc564fbad..d636699091b 100644
--- a/pyomo/contrib/piecewise/tests/test_piecewise_linear_function.py
+++ b/pyomo/contrib/piecewise/tests/test_piecewise_linear_function.py
@@ -416,6 +416,72 @@ def g2(x1, x2):
         self.assertAlmostEqual(m.pw(0.2, 4.3), g2(0.2, 4.3))
 
 
+class TestConvexOption(unittest.TestCase):
+    def test_scalar_pw_convex_option(self):
+        """Test that convex option is correctly set for ScalarPiecewiseLinearFunction"""
+        m = ConcreteModel()
+        m.x = Var(bounds=(1, 10))
+
+        # Test convex=True
+        m.pw_convex = PiecewiseLinearFunction(
+            points=[1, 3, 6, 10], function=log, convex=True
+        )
+        self.assertTrue(m.pw_convex.convex)
+
+        # Test convex=False
+        m.pw_concave = PiecewiseLinearFunction(
+            points=[1, 3, 6, 10], function=log, convex=False
+        )
+        self.assertFalse(m.pw_concave.convex)
+
+        # Test convex=None (default)
+        m.pw_none = PiecewiseLinearFunction(points=[1, 3, 6, 10], function=log)
+        self.assertIsNone(m.pw_none.convex)
+
+    def test_indexed_pw_convex_option_points(self):
+        """Test that convex option is correctly set for
+        IndexedPiecewiseLinearFunction"""
+        m = ConcreteModel()
+        m.x = Var([1, 2], bounds=(1, 10))
+
+        def g1(x):
+            return x**2
+
+        def g2(x):
+            return log(x)
+
+        m.funcs = {1: g1, 2: g2}
+
+        # True and False
+        m.pw_convex = PiecewiseLinearFunction(
+            [1, 2],
+            points=[1, 3, 6, 10],
+            function_rule=m.funcs,
+            convex={1: True, 2: False},
+        )
+        self.assertTrue(m.pw_convex[1].convex)
+        self.assertFalse(m.pw_convex[2].convex)
+
+        def convex_rule(m, i):
+            if i == 1:
+                return True
+            return None
+
+        # Test convex=False for both indices
+        m.pw_concave = PiecewiseLinearFunction(
+            [1, 2], points=[1, 3, 6, 10], function_rule=m.funcs, convex=convex_rule
+        )
+        self.assertTrue(m.pw_concave[1].convex)
+        self.assertIsNone(m.pw_concave[2].convex)
+
+        # Test convex=None (default) for both indices
+        m.pw_none = PiecewiseLinearFunction(
+            [1, 2], points=[1, 3, 6, 10], function_rule=lambda m, i: m.funcs[i]
+        )
+        self.assertIsNone(m.pw_none[1].convex)
+        self.assertIsNone(m.pw_none[2].convex)
+
+
 class TestTriangulationProducesDegenerateSimplices(unittest.TestCase):
     cube_extreme_pt_indices = [
         {10, 11, 13, 14, 19, 20, 22, 23},  # right bottom back
diff --git a/pyomo/contrib/piecewise/transform/epigraph_hypograph.py b/pyomo/contrib/piecewise/transform/epigraph_hypograph.py
new file mode 100644
index 00000000000..5d1845d56c8
--- /dev/null
+++ b/pyomo/contrib/piecewise/transform/epigraph_hypograph.py
@@ -0,0 +1,70 @@
+# ____________________________________________________________________________________
+#
+# Pyomo: Python Optimization Modeling Objects
+# Copyright (c) 2008-2026 National Technology and Engineering Solutions of Sandia, LLC
+# Under the terms of Contract DE-NA0003525 with National Technology and Engineering
+# Solutions of Sandia, LLC, the U.S. Government retains certain rights in this
+# software.  This software is distributed under the 3-clause BSD License.
+# ____________________________________________________________________________________
+
+from pyomo.common.config import ConfigDict, ConfigValue
+from pyomo.core.base import TransformationFactory
+from pyomo.contrib.piecewise.transform.piecewise_linear_transformation_base import (
+    PiecewiseLinearTransformationBase,
+)
+from pyomo.contrib.piecewise.transform.piecewise_to_mip_visitor import (
+    PiecewiseLinearToMIP,
+)
+from pyomo.environ import Constraint, NonNegativeIntegers, Var
+
+
+@TransformationFactory.register(
+    'contrib.piecewise.epigraph_hypograph',
+    doc="Transforms convex/concave piecewise linear functions to their epigraphical/"
+    "hypographical LP formulations",
+)
+class PWLToEpigraphOrHypograph(PiecewiseLinearTransformationBase):
+    CONFIG = PiecewiseLinearTransformationBase.CONFIG()
+    _transformation_name = 'pw_linear_epigraph'
+
+    def _transform_pw_linear_expr(self, pw_expr, pw_linear_func, transformation_block):
+        transBlock = transformation_block.transformed_functions[
+            len(transformation_block.transformed_functions)
+        ]
+        # get the PiecewiseLinearFunctionExpression
+        dimension = pw_expr.nargs()
+
+        # Create variable to substitute
+        substitute_var = transBlock.substitute_var = Var()
+        pw_linear_func.map_transformation_var(pw_expr, substitute_var)
+
+        transBlock.epigraphical_constraints = Constraint(NonNegativeIntegers)
+
+        epigraph = False
+        if pw_linear_func.convex is None:
+            # we should autodetect if the dimension isn't insane, yell otherwise
+            raise NotImplementedError(
+                "It is (quadratically, in the number of pieces) possible to "
+                "auto-detect if a piecewise-linear "
+                "function is convex or concave, but we don't currently have an "
+                "implementation. PRs are welcome, or manually specify in the "
+                "PiecewiseLinearFunction constructor the convexity/concavity "
+                "using the 'convex' argument."
+            )
+        if pw_linear_func.convex:
+            # This will be epigraph
+            epigraph = True
+        # Else this is the hypograph (the function is concave)
+
+        for idx, linear_func in enumerate(pw_linear_func._linear_functions):
+            linear_func_expr = linear_func(*pw_expr.args)
+            if epigraph:
+                transBlock.epigraphical_constraints[idx] = (
+                    linear_func_expr <= substitute_var
+                )
+            else:
+                transBlock.epigraphical_constraints[idx] = (
+                    linear_func_expr >= substitute_var
+                )
+
+        return substitute_var