Kenris Illustrative examples are presented to demonstrate the correctness of proposed model. There is no such single optimal solution exist which could optimize all the objective functions simultaneously. The basic concept and classical principle of multi-objective optimization problems with KKT condition has been discussed. Notes will be shown in their authored language.

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For alternatives, see Optimization App Alternatives. Start the Optimization app by entering optimtool at the command line. For more information about this tool, see Optimization App. The default Solver, fmincon - Constrained nonlinear minimization, is selected. For more information about choosing a solver, see Optimization Decision Table.

The default Algorithm, Interior point, is also selected. In the Objective function box, enter rosenbrock. In the Start point box, enter [0 0] to specify the initial point where fmincon begins its search for a minimum. In the Nonlinear constraint functionbox, enter unitdisk, the function handle of unitdisk. Ensure that your Problem Setup and Results pane matches this figure. In the Options pane, under Display to command window at the bottom of the pane , select iterative from the Level of display list.

If you do not see the option, click Display to command window. This setting shows the progress of fmincon in the command window. The following message appears in the Run solver and view results box: Optimization running. Objective function value: 0. 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.

Your objective function value can differ slightly, depending on your computer system and version of Optimization Toolbox. The message tells you that: The search for a constrained optimum ended because the derivative of the objective function is nearly 0 in directions allowed by the constraint.

The constraint is satisfied to the requisite accuracy. At the bottom of the Problem Setup and Results pane, the minimizer x appears under Final point. For more information about exit messages, see Exit Flags and Exit Messages. Create options that choose iterative display and the interior-point algorithm.

See the fmincon function reference pages for the syntax. Local minimum found that satisfies the constraints. Several phrases in the message contain links to more information about the terms used in the message.

For more details about these links, see Enhanced Exit Messages. This table is the same whether you use the Optimization app or the command line. The following description applies to the table as displayed. The first column, labeled Iter, is the iteration number from 0 to The third column, labeled f x , displays the value of the objective function.

The final value, 0. The fourth column, Feasibility, is 0 for all iterations. This column shows the value of the constraint function unitdisk at each iteration where the constraint is positive.

Because the value of unitdisk was negative in all iterations, every iteration satisfied the constraint. The other columns of the iteration table are described in Iterative Display.

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