بهینه‌سازی

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مقدمه

بهینه‌‌‌‌سازی یکی از مفاهیم پرکاربرد در شاخه‌های مختلف علوم و مهندسی می‌باشد که در بسیاری از مسائل روزمره، از جمله جستجو در اینترنت، برنامه‌ریزی‌های اقتصادی و بسیاری از مسائل دیگر با آن برخورد می‌‌کنیم. در مسائل بهینه‌سازی، معمولاً هدف، رسیدن به مقادیری از پارامترها برای کمینه کردن و یا بیشینه کردن یک مجموعه از توابع هدف می‌باشد. منابعی نظیر زمان و پول در دنیای واقعی، منابعی محدود بوده و باید روشی بهینه برای حداکثر استفاده از آن‌ها یافت. راه‌‌های معمول حل‌ مسائل بهینه‌سازی شامل روش‌‌های عددی و الگوریتم‌های احتمالاتی می‌‌‌باشند که به‌عنوان ابزاری مفید برای مسائل بهینه‌سازی مورد استفاده قرار می‌‌‌گیرند.

تعریف ریاضی بهینه‌سازی

از نظر مفهومی می‌توان یک مساله‌ی بهینه‌سازی را به‌صورت کمینه‌ و یا بیشنه کردن یک تابع مانند ، در زیر مجموعه‌ای از دامنه‌اش در نظر گرفت. این دو مساله به سادگی با تغییر علامت مقادیر خروجی تابع مورد نظر، قابل تبدیل به هم هستند. از نظر ریاضی می‌توان مسائل بهینه‌سازی برای کمینه‌سازی یک تابع را به شکل‌ رسمی زیر نوشت:

\begin{align} minimize ~f_i(x),~x\in \Re^n and~ i=1, 2,…,I \end{align}

با داشتن شرایط زیر

\begin{align} &h_j(x)=0,~(j=1, 2,…,J),
&g_k(x) \leq0,~(k=1, 2,…,K). \end{align}

که توابع ، و معرف سه تابع هستند که بردار ورودی ‌آن‌ها می‌باشد و دو تابع آخر قیود اعمال‌شده روی می‌باشند. تابع را در مسائل بهینه‌سازی، تابع هدف می‌نامند. در صورتی که باشد، مساله بهینه‌سازی تک‌هدفه و در غیر این صورت چند‌هدفه نام دارد. فصای جستجو مساله می‌باشد که درون آن به‌دنبال جواب هستیم و فضای حاصل از مقادیر خروجی توابع هدف را فضای جواب می‌نامیم. عبارات
و قیود مساله بهینه‌سازی هستند که می‌خواهیم توابع هدف را تحت این قیود کمینه کنیم. در تعدادی از مسائل، تابع هدفی وجود ندارد و فقط با قیود روبرو هستیم در این‌گونه مسائل، هر ترکیبی از قیود می‌تواند جواب مورد نظر باشد.

توجه

دسته‌بندی روش‌های محاسباتی به دلیل تنوع این روش‌ها و ماهیت مساله‌هایی که می‌توان هر روش‌ را در آن اعمال نمود بسیار متنوع است. برای همین در ادامه با آوردن مثال‌هایی از مساله‌های مختلف بهینه‌سازی به بررسی ويژگی‌های این روش‌ها و شرایط کاربرد آن‌ها خواهیم پرداخت.

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