I am building an multi-objective optimization model with a quadratic constraint. However, when I solve the model I get the following error :

CPLEX Error 1031: Not available for QCP

When I replace multi-objective function with the single objective objective function, I get the following error :

CPLEX Error 5002: 'q1' is not convex.

I am enclosing the code. There is one quadratic constraint that gets introduced as I need to calculate the average margin (total margin / number of items). Now, both numerator and denominator expression have decision variables, I have used logical constraints (using indicator variables to get my way through). Is there any other way of calculating average, without making the constraint quadratic ?

mdl = Model('assortment_optimization')
sku_binary_var = mdl.binary_var_dict(skus, lb = 0, ub = 1, name = lambda p: '%s' % p)

num_sku_selected = mdl.integer_var(name = 'total_selected_sku')
mdl.add_constraint(num_sku_selected == mdl.sum(sku_binary_var[i] for i in skus))

mdl.add_constraint(mdl.sum(sku_binary_var[i] * sku_char[i]['shelf_capacity'] * sku_char[i]['item_cbm'] for i in skus) <= VOLUME_CAPACITY_CBM, ctname = 'volume_constraint')

mdl.add_constraint(mdl.sum(sku_binary_var[i] * sku_char[i]['shelf_capacity'] * sku_char[i]['cost'] for i in skus) <= MAX_COST, ctname = 'cost_constraint')

# mdl.add_constraint(mdl.sum(sku_binary_var[i] for i in skus) <= 233)

tot_margin = mdl.continuous_var(name = 'tot_margin')
avg_margin = mdl.continuous_var(name = 'avg_margin')

mdl.add_constraint(tot_margin == mdl.sum(sku_binary_var[i] * sku_char[i]['avg_wk_sales_units'] * sku_char[i]['margin'] for i in skus))

# indicator constraints to help calculate average margin
indic_varbs = mdl.binary_var_dict(range(1, len(skus)+1))

for i in range(1, len(skus)+1):
    mdl.add_indicator(indic_varbs[i], num_sku_selected == i, active_value = 1)

mdl.add_constraint(mdl.sum(indic_varbs[i] for i in range(1, len(skus)+1)) == 1)

# this becomes a quadratic constraint
mdl.add_constraint(avg_margin == tot_margin * mdl.sum(indic_varbs[i] * (1 / i) for i in range(1, len(skus) + 1)))

mdl.add_constraint(avg_margin >= 8)

mdl.add_kpi(num_sku_selected, 'Number of SKUs in the new assortment')

# below are the objectives that I'd like to maximize

obj_margin = mdl.sum(sku_binary_var[i] * sku_char[i]['avg_wk_sales_units'] * sku_char[i]['margin'] for i in skus)
obj_freq = mdl.sum(sku_binary_var[i] * sku_char[i]['freq_trans'] for i in skus)
obj_unique = mdl.sum(sku_binary_var[i] * sku_char[i]['uniq_factor'] for i in skus)

# mdl.maximize(obj_margin)

mdl.maximize_static_lex(exprs=[obj_margin, obj_freq, obj_unique])

mdl.solve(log_output=True)

------------------------------
Bhartendu Awasthi
------------------------------

#DecisionOptimization

If T, A and x_i are respectively the total margin, average margin and indicators for number of items, then T = A * sum_i x_i = sum_i (A * x_i). That is still quadratic (and nonconvex), but assuming you know bounds on the average margin you can now linearize each of the products A * x_i using some additional continuous variables and some additional constraints (https://orinanobworld.blogspot.com/2010/10/binary-variables-and-quadratic-terms.html).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University
------------------------------

Thank you Dr Rubin for the ingenious work around. I certainly can linearize the product of a continuous and a binary variable. 

Just had one general follow up question - in case the solver (i know some solvers have the capability to solve Non convex QCP) has the capability to handle quadratic constraints (non - convex), then is there a merit in linearizing ? Would the solver speed be higher if we linearize the constraints. Pardon me if this question does not make much sense, I am relatively new to the world of mathematical programming.

Thanks again,

------------------------------
Bhartendu Awasthi
------------------------------

Original Message:
Sent: Sat June 12, 2021 02:15 PM
From: Paul Rubin
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

If T, A and x_i are respectively the total margin, average margin and indicators for number of items, then T = A * sum_i x_i = sum_i (A * x_i). That is still quadratic (and nonconvex), but assuming you know bounds on the average margin you can now linearize each of the products A * x_i using some additional continuous variables and some additional constraints (https://orinanobworld.blogspot.com/2010/10/binary-variables-and-quadratic-terms.html).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University
------------------------------

If the solver "handles" non-convex quadratic constraints, the question becomes what "handle" means. If the solver guarantees a globally optimal solution, then the issue reduces to speed, and I honestly do not know which would be faster. I'm not sure how many solvers guarantee global optimality versus local optimality in this case. Linearizing results in a guaranteed optimal solution (assuming you have enough time and memory, which is itself not certain).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University
------------------------------

Original Message:
Sent: Sun June 13, 2021 12:37 AM
From: Bhartendu Awasthi
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

Thank you Dr Rubin for the ingenious work around. I certainly can linearize the product of a continuous and a binary variable. 

Just had one general follow up question - in case the solver (i know some solvers have the capability to solve Non convex QCP) has the capability to handle quadratic constraints (non - convex), then is there a merit in linearizing ? Would the solver speed be higher if we linearize the constraints. Pardon me if this question does not make much sense, I am relatively new to the world of mathematical programming.

Thanks again,

------------------------------
Bhartendu Awasthi
------------------------------


Original Message:
Sent: Sat June 12, 2021 02:15 PM
From: Paul Rubin
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

If T, A and x_i are respectively the total margin, average margin and indicators for number of items, then T = A * sum_i x_i = sum_i (A * x_i). That is still quadratic (and nonconvex), but assuming you know bounds on the average margin you can now linearize each of the products A * x_i using some additional continuous variables and some additional constraints (https://orinanobworld.blogspot.com/2010/10/binary-variables-and-quadratic-terms.html).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University

Thank you very much Dr Rubin.

Regards,
Bhartendu

------------------------------
Bhartendu Awasthi
------------------------------

Original Message:
Sent: Sun June 13, 2021 12:12 PM
From: Paul Rubin
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

If the solver "handles" non-convex quadratic constraints, the question becomes what "handle" means. If the solver guarantees a globally optimal solution, then the issue reduces to speed, and I honestly do not know which would be faster. I'm not sure how many solvers guarantee global optimality versus local optimality in this case. Linearizing results in a guaranteed optimal solution (assuming you have enough time and memory, which is itself not certain).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University
------------------------------


Original Message:
Sent: Sun June 13, 2021 12:37 AM
From: Bhartendu Awasthi
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

Thank you Dr Rubin for the ingenious work around. I certainly can linearize the product of a continuous and a binary variable. 

Just had one general follow up question - in case the solver (i know some solvers have the capability to solve Non convex QCP) has the capability to handle quadratic constraints (non - convex), then is there a merit in linearizing ? Would the solver speed be higher if we linearize the constraints. Pardon me if this question does not make much sense, I am relatively new to the world of mathematical programming.

Thanks again,

------------------------------
Bhartendu Awasthi


Original Message:
Sent: Sat June 12, 2021 02:15 PM
From: Paul Rubin
Subject: Using quadratic constraint(s) with multi-objective function (docplex)

If T, A and x_i are respectively the total margin, average margin and indicators for number of items, then T = A * sum_i x_i = sum_i (A * x_i). That is still quadratic (and nonconvex), but assuming you know bounds on the average margin you can now linearize each of the products A * x_i using some additional continuous variables and some additional constraints (https://orinanobworld.blogspot.com/2010/10/binary-variables-and-quadratic-terms.html).

------------------------------
Paul Rubin
Professor Emeritus
Michigan State University