The following describes constrained optimization problems more precisely, restricting the discussion to minimization problems for brevity.
A local minimum may not be a global minimum. A global minimum is always a local minimum.
Wolfram Language functions for constrained optimization include Minimize , Maximize , NMinimize , and NMaximize for global constrained optimization, FindMinimum for local constrained optimization, and LinearOptimization for efficient and direct access to linear optimization methods. The following table briefly summarizes each of the functions.
, | numeric local optimization | linear programming methods, nonlinear interior point algorithms, utilize second derivatives |
, | numeric global optimization | linear programming methods, Nelder-Mead, differential evolution, simulated annealing, random search |
, | exact global optimization | linear programming methods, cylindrical algebraic decomposition, Lagrange multipliers and other analytic methods, integer linear programming |
linear optimization | linear optimization methods (simplex, revised simplex, interior point) |
Summary of constrained optimization functions.
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Mathematical tools for intermediate economics classes Iftekher Hossain
Calculus of Multivariable Functions
Although there are examples of unconstrained optimizations in economics, for example finding the optimal profit, maximum revenue, minimum cost, etc., constrained optimization is one of the fundamental tools in economics and in real life. Consumers maximize their utility subject to many constraints, and one significant constraint is their budget constraint. Even Bill Gates cannot consume everything in the world and everything he wants. Can Mark Zuckerberg buy everything? Similarly, while maximizing profit or minimizing costs, the producers face several economic constraints in real life, for examples, resource constraints, production constraints, etc.
The commonly used mathematical technique of constrained optimizations involves the use of Lagrange multiplier and Lagrange function to solve these problems followed by checking the second order conditions using the Bordered Hessian. When the objective function is a function of two variables, and there is only one equality constraint , the constrained optimization problem can also be solved using the geometric approach discussed earlier given that the optimum point is an interior optimum. It should be mentioned again that we will not address the second-order sufficient conditions in this chapter.
Example 1: Maximize utility \(u = f(x,y) = xy\) subject to the constraint \(g(x,y) = x + 4y = 240\). Here the price of per unit \(x\) is \(1\), the price of \(y\) is \(4\) and the budget available to buy \(x\) and \(y\) is \(240\). Solve the problem using the geometric approach .
Here the optimization problem is: Objective function: maximize \(u(x,y) = xy\) Subject to the constraint: \(g(x,y) = x + 4y = 240\). Step 1: \(-\frac{f_{x}}{f_{y}} = -\frac{y}{x}\) (Slope of the indifference curve) Step 2: \(-\frac{g_{x}}{g_{y}} = -\frac{1}{4}\) (Slope of the budget line) Step 3: \(-\frac{f_{x}}{f_{y}} = -\frac{g_{x}}{g_{y}}\) (Utility maximization requires the slope of the indifference curve to be equal to the slope of the budget line.) $$-\frac{y}{x} = -\frac{1}{4}$$ $$x = 4y$$ Step 4: From step 3, use the relation between \(x\) and \(y\) in the constraint function to get the critical values. $$x + 4y = 240$$ $$4y + 4y = 240$$ $$8y = 240$$ $$y = 30$$ Using \(y = 30\) in the relation \(x = 4y\), we get \(x = 4 \times 30 = 120\) Utility may be maximized at \((120, 30)\).
Suppose a consumer consumes two goods, \(x\) and \(y\) and has utility function \(u(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = 10\) and the price of \(y\) is \(P_{y} = 20\). Find his optimal consumption bundle using the Lagrange method .
Here the optimization problem is: Objective function: maximize \(u(x,y) = xy\) Subject to the constraint: \(g(x,y) = 10x + 20y = 400\). This is a problem of constrained optimization. Form the Lagrange function: $$L(x,y,\mu ) \equiv \color{red}{f(x,y)} - \mu (\color{purple}{g(x,y) - k})$$ $$L(x,y,\mu ) \equiv xy - \mu (10x + 20y - 400)$$ Set each first order partial derivative equal to zero: $$\frac{\partial L}{\partial x} = y - 10\mu = 0 \qquad\qquad\qquad \text{(1)}$$ $$\frac{\partial L}{\partial y} = x - 20\mu = 0 \qquad\qquad\qquad \text{(2)}$$ $$\frac{\partial L}{\partial \mu} = -(10x + 20y - 400) = 0 \quad \text{(1)}$$ From equations (1) and (2) we find: $$x = 2y$$ Use \(x = 2y\) in equation (3) to get: $$10x + 20y = 400$$ $$40y = 400$$ $$\bf{y = 10}$$ $$\bf{x = 2y = 20}$$ See the graph below.
Suppose a consumer consumes two goods, \(x\) and \(y\) and has the utility function \(U(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = 10\) and the price of \(y\) is \(P_{y} = 20\).
Suppose a consumer consumes two goods, \(x\) and \(y\) and has utility function \(U(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = $10\) and the price of \(y\) is \(P_{y} = $20\).
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Typical optimization problem.
This example shows how to solve a constrained nonlinear optimization problem using the problem-based approach. The example demonstrates the typical work flow: create an objective function, create constraints, solve the problem, and examine the results.
If your objective function or nonlinear constraints are not composed of elementary functions, you must convert the nonlinear functions to optimization expressions using fcn2optimexpr . See the last part of this example, Alternative Formulation Using fcn2optimexpr , or Convert Nonlinear Function to Optimization Expression .
For the solver-based approach to this problem, see Constrained Nonlinear Problem Using Optimize Live Editor Task or Solver .
Consider the problem of minimizing the logarithm of 1 plus Rosenbrock's function
f ( x ) = log ( 1 + 1 0 0 ( x 2 - x 1 2 ) 2 + ( 1 - x 1 ) 2 ) ,
over the unit disk , meaning the disk of radius 1 centered at the origin. In other words, find x that minimizes the function f ( x ) over the set x 1 2 + x 2 2 ≤ 1 . This problem is a minimization of a nonlinear function subject to a nonlinear constraint.
Rosenbrock's function is a standard test function in optimization. It has a unique minimum value of 0 attained at the point [1,1] , and therefore f ( x ) attains the same minimum at the same point. Finding the minimum is a challenge for some algorithms because the function has a shallow minimum inside a deeply curved valley. The solution for this problem is not at the point [1,1] because that point does not satisfy the constraint.
This figure shows two views of the function f ( x ) function in the unit disk. Contour lines lie beneath the surface plot.
The rosenbrock function handle calculates the function f ( x ) at any number of 2-D points at once. This Vectorization speeds the plotting of the function, and can be useful in other contexts for speeding evaluation of a function at multiple points.
The function f ( x ) is called the objective function. The objective function is the function you want to minimize. The inequality x 1 2 + x 2 2 ≤ 1 is called a constraint. Constraints limit the set of x over which a solver searches for a minimum. You can have any number of constraints, which are inequalities or equations.
The problem-based approach to optimization uses optimization variables to define objective and constraints. There are two approaches for creating expressions using these variables:
For elementary functions such as polynomials or trigonometric functions, write expressions directly in the variables.
For other types of functions, convert functions to optimization expressions using fcn2optimexpr . See Alternative Formulation Using fcn2optimexpr at the end of this example.
For this problem, both the objective function and the nonlinear constraint are elementary, so you can write the expressions directly in terms of optimization variables. Create a 2-D optimization variable named 'x' .
Create the objective function as an expression of the optimization variable.
Create an optimization problem named prob having obj as the objective function.
Create the nonlinear constraint as a polynomial of the optimization variable.
Include the nonlinear constraint in the problem.
Review the problem.
To solve the optimization problem, call solve . The problem needs an initial point, which is a structure giving the initial value of the optimization variable. Create the initial point structure x0 having an x -value of [0 0] .
The solution shows exitflag = OptimalSolution . This exit flag indicates that the solution is a local optimum. For information on trying to find a better solution, see When the Solver Succeeds .
The exit message indicates that the solution satisfies the constraints. You can check that the solution is indeed feasible in several ways.
Check the reported infeasibility in the constrviolation field of the output structure.
An infeasibility of 0 indicates that the solution is feasible.
Compute the infeasibility at the solution.
Again, an infeasibility of 0 indicates that the solution is feasible.
Compute the norm of x to ensure that it is less than or equal to 1.
The output structure gives more information on the solution process, such as the number of iterations (23), the solver ( fmincon ), and the number of function evaluations (44). For more information on these statistics, see Tolerances and Stopping Criteria .
For more complex expressions, write function files for the objective or constraint functions, and convert them to optimization expressions using fcn2optimexpr . For example, the basis of the nonlinear constraint function is in the disk.m file:
Convert this function file to an optimization expression.
Furthermore, you can also convert the rosenbrock function handle, which was defined at the beginning of the plotting routine, into an optimization expression.
Create an optimization problem using these converted optimization expressions.
View the new problem.
Solve the new problem. The solution is essentially the same as before.
For the list of supported functions, see Supported Operations for Optimization Variables and Expressions .
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Bibliometrics & citations, view options, recommendations, interior gradient and epsilon-subgradient descent methods for constrained convex minimization.
We extend epsilon-subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets, in particular over R p +. This is done by replacing the usual squared quadratic regularization term used in ...
In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the ...
We study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value. We show that they enjoy almost optimal efficiency estimates. We ...
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Title: a fast single-loop primal-dual algorithm for non-convex functional constrained optimization.
Abstract: Non-convex functional constrained optimization problems have gained substantial attention in machine learning and signal processing. This paper develops a new primal-dual algorithm for solving this class of problems. The algorithm is based on a novel form of the Lagrangian function, termed {\em Proximal-Perturbed Augmented Lagrangian}, which enables us to develop an efficient and simple first-order algorithm that converges to a stationary solution under mild conditions. Our method has several key features of differentiation over existing augmented Lagrangian-based methods: (i) it is a single-loop algorithm that does not require the continuous adjustment of the penalty parameter to infinity; (ii) it can achieves an improved iteration complexity of $\widetilde{\mathcal{O}}(1/\epsilon^2)$ or at least ${\mathcal{O}}(1/\epsilon^{2/q})$ with $q \in (2/3,1)$ for computing an $\epsilon$-approximate stationary solution, compared to the best-known complexity of $\mathcal{O}(1/\epsilon^3)$; and (iii) it effectively handles functional constraints for feasibility guarantees with fixed parameters, without imposing boundedness assumptions on the dual iterates and the penalty parameters. We validate the effectiveness of our method through numerical experiments on popular non-convex problems.
Subjects: | Optimization and Control (math.OC) |
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We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method.
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The authors are grateful to the two referees for their helpful comments and constructive suggestions. In particular, we thank one of the referees for suggesting the use of weaker qualifications than MPEC-RCPLDQC when studying the convergence. The first author was supported by the National Natural Science Foundation of China (Grants 72131007, 72140006, 12271161) and the Natural Science Foundation of Shanghai (Grant 22ZR1415900). This second author was supported by the Project of National Center for Applied Mathematics (Grant ncamc2021-msxm01) and the National Natural Science Foundation of China (Grant 11901068).
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When using the regularization scheme by Kadrani et al., our approximation problem becomes
where \(\Phi _i^{KDB}(x;t):=(H_i(x)-t)(G_i(x)-t)\) for all \(i\in \{1,\ldots ,m\}\) . By a straightforward calculation, we have
Assume that \(t^{k}\downarrow 0\) and \(\sigma ^{k}\downarrow 0\) as \(k\rightarrow \infty \) . Let \(x^k\) be a stationary solution of \(\mathrm{(P_{\sigma ^k,t^k}^{KDB})}\) , and \(x^*\) be an accumulation point of \(\{x^k\}\) . If MPEC-CCQC holds at \(x^*\) , then \(x^*\) is an M-stationary point of problem ( 1 ).
Recall that \(x^k\) is a stationary point of problem \(\mathrm{(P_{\sigma ^k,t^k}^{KDB})}\) if there exist multipliers \((\alpha ^k,\beta ^k,\pi ^k,\rho ^k,\gamma ^k)\) such that
Then ( 36 ) can be rewritten as
where \(\xi ^k_i \in \partial \varphi (D_i^\top x^k; \sigma ^k_i)\) for all \(i=1,\ldots ,r\) . By the definition of \(\varphi \) , it is easy to see that
We next show that there exists a sequence \(z^k\rightarrow F(x^*)\) such that ( 41 ) holds with \(\alpha _i^k\in {{\mathcal {N}}}_{(0,\infty ]}(z_i^k)\) and \((\mu _i^k,\nu _i^k)\in {{\mathcal {N}}}_{{\mathcal {C}}}(z_i^k)\) . Let \(z^k = (D_i^Tx^*\ i\in {{\mathcal {I}}}_0^*,g(x^k),h(x^k),R(x^*))\) . It is clear that \(z^k\in \Lambda \) . By ( 37 ), it is easy to see that \(\alpha _i^k\in {{\mathcal {N}}}_{(0,\infty ]}(z_i^k)\) for all \(i=1,\ldots ,q\) . One can easily observe that if \(i\in {{\mathcal {I}}}_{+0}^*\) , then \(G_i(x^*)>0\) . Thus \(\pi _i^k=0\) by ( 38 ) and \(\gamma _i^k(H_i(x^k)-t_k)=0\) by ( 40 ). Hence \(\mu _i^k=\pi _i^k-\gamma _i^k(H_i(x^k)-t_k)=0\) for \(i\in {{\mathcal {I}}}_{+0}^*\) . In the same way, we have \(\nu _i^k=0\) for \(i\in {{\mathcal {I}}}_{0+}^*\) . Moreover, it is easy to see that if \(\mu _i^k<0\) , then \(\pi _i^k=0\) and \(\gamma _i^k(H_i(x^k)-t_k)>0\) . Thus, by ( 40 ), it follows \(G_i(x^k)-t_k = 0\) and by ( 39 ), it follows \(\rho _i^k=0\) . Hence \(\nu _i^k=\rho _i^k-\gamma _i^k(G_i(x^k)-t_k)=0\) and then \(\mu _i^k\nu _i^k=0\) if \(\mu _i^k<0\) . In the same way, we can have \(\mu _i^k\nu _i^k=0\) if \(\nu _i^k<0\) . Consequently, we have \(\mu _i^k>0,\nu _i^k>0\) or \(\mu _i^k\nu _i^k=0\) for all i . Thus, by the expression of the normal cone to \({{\mathcal {C}}}\) , we have \((\mu _i^k,\nu _i^k)\in {{\mathcal {N}}}_{{\mathcal {C}}}(z_i^k)\) .
Based on the above discussions and the expression of the normal cone to \(\Lambda \) , ( 41 ) can be written as
Since MPEC-CCQC holds at \(x^*\) , it follows that
indicating that \(x^*\) is an M-stationary point by Proposition 2 . \(\square \)
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Guo, L., Li, G. Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02475-6
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DOI : https://doi.org/10.1007/s10957-024-02475-6
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Luckily there are many numerical methods for solving constrained optimization problems, though we will not discuss them here. This page titled 2.7: Constrained Optimization - Lagrange Multipliers is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Michael Corral via source content that was ...
The constrained-optimization problem (COP) is a significant generalization of the classic constraint-satisfaction problem (CSP) ... Virtually, this corresponds on ignoring the evaluated variables and solving the problem on the unassigned ones, except that the latter problem has already been solved. More precisely, the cost of soft constraints ...
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However, solving constrained optimization problems is a very important topic in applied mathematics. The techniques developed here are the basis for solving larger problems, where more than two variables are involved. We illustrate the technique again with a classic problem.
The "Lagrange multipliers" technique is a way to solve constrained optimization problems. Super useful! Background. Contour maps; ... However, if you (or more realistically a computer) were solving a given constrained optimization problem, it's not like you would first find the unconstrained maximum, check if it fits the constraint, then turn ...
Constrained optimization introduction. The Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem occurs when the contour lines of the function being maximized are tangent to the constraint curve. Created by Grant Sanderson.
Practice Problem 1 1. Write constraints for each of the following: a) A batch of cookies requires 3 cups of flour, and a cake requires 4. Write a constraint limiting the amount of cookies and cakes that can be made with 24 cups of flour. b) Box type 1 can hold 20 books and box type 2 can hold 12. Write a constraint for the number of boxes
If you find yourself solving a constrained optimization problem by hand, and you remember the idea of gradient alignment, feel free to go for it without worrying about the Lagrangian. In practice, it's often a computer solving these problems, not a human.
2 to get. solving this is using 1 = x21 + (2x2)2 4x1x2 wherep pt. e equality holds when x1 = 2x2.So x1 = 2=2 and x2 = 2=4.However, not all equality-constrained. We need general theory to solve constrained optimization problems with equal-ity constraints: mize f (x) subject t. h(x) = 0and the fe.
onA constrained optimization problem is a problem of the formmaximize (or minimi. (x, y) subject to the condition g(x, y) = 0.1From two to oneIn some cases one can solve for y as a func. ion of x and then find the extrema of a one variable function.That is, if the equation g(x, y) = 0 is equivalent to y = h(x), then we may set f(x) = F (x, h(x)) a.
maximize L, once has been chosen. This provides an intuition into this method of solving the constrained maximization problem. In the constrained problem we have told the decision maker that she must satisfy g(x 1;:::;x n) = c and that she should choose among all points that satisfy this constraint the point at which f(x 1;:::;x n) is greatest ...
Optimization III: Constrained Optimization. Really Di cult! \Maximize pro ts where you make a positive amount of each product and use limited material." A feasible point is any point ~x satisfying g(~x) = ~ 0 and h(~x) ~ 0: The feasible set is the set of all points ~x satisfying these constraints. A feasible point is any point ~x satisfying g ...
It follows from this that we can solve constrained optimization problems with equality constraints as follows (assuming ∇ᵤg(u*)≠0): ... this method of Lagrange multipliers is powerful in that it casts a constrained optimization problem into an unconstrained optimization problem which we can solve by simply setting the gradient as zero.
Theorem 5.1 Suppose that f(x) is twice differentiable on the open convex set S. Then f(x) is a convex function on the domain S if and only if H(x) is SPSD for all x S. ∈. The following functions are examples of convex functions in n-dimensions. f(x) = aT x + b. f(x) = 1xT Mx cT x where M is SPSD. 2 −. f(x) = x.
Constrained optimization problems are problems for which a function f(x) is to be minimized or maximized subject to constraints \[CapitalPhi] (x). Here f:\[DoubleStruckCapitalR]^n-> \[DoubleStruckCapitalR] is called the objective function and \[CapitalPhi](x) is a Boolean-valued formula. ... Solving Optimization Problems. The methods used to ...
Constrained Optimization Definition. Constrained minimization is the problem of finding a vector x that is a local minimum to a scalar function f ( x ) subject to constraints on the allowable x: min x f ( x) such that one or more of the following holds: c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u. There are even more ...
Photo by Drew Dizzy Graham on Unsplash. Interior Point Methods typically solve the constrained convex optimization problem by applying Newton Method to a sequence of equality constrained problems. Barrier methods, as the name suggest, employ barrier functions to integrate inequality constraints into the objective function. Since we want to merge inequality constraints to the objective, the ...
Lagrange multipliers, using tangency to solve constrained optimization. The Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem occurs when the contour lines of the function being maximized are tangent to the constraint curve.
Constrained Optimization and Lagrange Multipliers. In Preview Activity 10.8.1, we considered an optimization problem where there is an external constraint on the variables, namely that the girth plus the length of the package cannot exceed 108 inches. We saw that we can create a function g from the constraint, specifically g(x, y) = 4x + y.
Solving problem using fmincon. Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
The commonly used mathematical technique of constrained optimizations involves the use of Lagrange multiplier and Lagrange function to solve these problems followed by checking the second order conditions using the Bordered Hessian. When the objective function is a function of two variables, and there is only one equality constraint, the ...
Constrained Optimization. Added Mar 16, 2017 by vik_31415 in Mathematics. Constrained Optimization. Send feedback | Visit Wolfram|Alpha. Get the free "Constrained Optimization" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.
Create an optimization problem named prob having obj as the objective function. prob = optimproblem( 'Objective' ,obj); Create the nonlinear constraint as a polynomial of the optimization variable. nlcons = x(1)^2 + x(2)^2 <= 1; Include the nonlinear constraint in the problem. prob.Constraints.circlecons = nlcons;
In recent years, the Cuckoo Optimization Algorithm (COA) has been widely used to solve various optimization problems due to its simplicity, efficacy, and capability to avoid getting trapped in local optima. However, COA has some limitations such as low convergence when it comes to solving constrained optimization problems with many constraints.
With the convex and smooth objective satisfying locally Lipschitz gradient we obtain the complexity O ( 1 k ) of f ( x k ) − f ⁎ at most. By using the idea of the new stepsize, we propose another new algorithm based on the projected gradient for solving a class of nonconvex optimization problems over a closed convex set.
Non-convex functional constrained optimization problems have gained substantial attention in machine learning and signal processing. This paper develops a new primal-dual algorithm for solving this class of problems. The algorithm is based on a novel form of the Lagrangian function, termed {\\em Proximal-Perturbed Augmented Lagrangian}, which enables us to develop an efficient and simple first ...
For any \(t \ne 0\), we let \(\textrm{sign}(t) = 1\) if \(t > 0\) and \(\textrm{sign}(t) = -1\) otherwise.. Problem is a nonconvex problem and it is necessary to investigate the convergence to its stationary points.For optimization problems with locally Lipschitz functions, a CQ such as the popular linear independence constraint qualification is required to ensure that local minimizers are KKT ...